I 


J) 


4  *?< 


SHORT  AND  COMPREHENSIVE  COURSE 


OF 


•  GEOMETRY  AND  TRIGONOMETRY; 


DESIGNED     FOR    GENERAL     USE     IN 


SCHOOLS    AND    COLLEGES. 


BY 


ANDREW  H.  BAKER,  A.M.,  Ph.D. 


NEW    YORK: 
P.     O'SHEA,     PUBLISHER, 

37     BARCLAY     STREET. 
I878. 


Copyright,  1878,  by  Patrick  O'Shea. 


Electrotyped  by  SMITH  &  McDOUGAL,  82  Beekman  St.,   New  York. 


PREFACE. 


GEOMETRY,  like  every  other  science,  has  but  few  principles, 
which,  if  systematically  arranged  and  thoroughly  developed, 
may  be  readily  comprehended,  and  indelibly  impressed  upon  the 
mind. 

Plane  Geometry  may  be  said  to  begin  and  end  with  the  circle. 
The  angles  formed  at  the  center  by  the  radii  should  be  treated  of 
first;  next,  inscribed  angles,  and  an  inscribed  triangle;  then,  a 
hexagon  and  an  equilateral  triangle ;  inscribed  and  circumscribed 
squares;  regular  and  irregular  polygons;  and  finally  return  to 
the  circle.  In  the  demonstrations  of  the  propositions  derived 
from  these  figures,  every  principle  of  Plane  Geometry  is  devel- 
oped ;  and  Solid  Geometry  has  no  distinct  principles. 

In  teaching  Solid  Geometry,  I  much  prefer  the  use  of  blocks. 
The  beginner,  at  least,  will  be  greatly  benefited  by  having  a 
material  object  to  inspect  and  compute,  until  he  becomes  thor- 
oughly acquainted  with  all  its  properties;  after  which,  he  may 
employ  his  imagination  as  he  likes,  and  conceive  figures  of  every 
shape  and  form.  The  system  of  object  teaching  favors  this 
method. 

In  preparing  this  treatise,  I  have  aimed  especially  at  simplicity 
and  brevity.  The  former,  that  it  may  be  within  the  grasp  of 
every  student;  the  latter,  that  the  memory  may  not  be  over- 
burdened, and  too  much  time  occupied  in  acquiring  a  thorough 
knowledge  of  the  science. 

Although  I  have  stated  that  beginners  derive  benefits  from 
material  figures,  I  do  not  thereby  wish  to  intimate  that  Geometry 


4  PEEFACE. 

presents  no  opportunity  for  the  exercise  of  the  imagination ; 
whilst,  in  truth,  no  other  science  presents  so  wide  a  field  for  the 
exercise  of  this  faculty.  We  pierce  the  most  distant  points  of  the 
celestial  concave  with  straight  lines,  and  with  arcs  of  great  cir- 
cles ;  measure  and  compute  the  distance  and  size  of  the  farthest 
stars,  thereby  rendering  what  appeared  imaginary,  matters  of 
fact. 

Simplicity  and  brevity  are  not  only  important,  but  they  are 
absolutely  necessary,  in  order  that  mankind  generally  may  acquire 
a  thorough  knowledge  of  pure  mathematics  ;  after  which,  any 
branch  of  applied  mathematics  may  be  pursued  with  ease  and 
advantage. 

The  design  of  the  Trigonometry  is  the  same  as  of  the  Geometry. 

"With  these  impressions,  I  dedicate  this  volume  to  the  Amer- 
ican Youth ;  and  if  it  prove  that  I  have  plucked  a  few  thorns 
from  the  rugged  path  of  science,  and  strewn  a  few  flowers  therein, 

I  shall  not  regret  the  arduous  task. 

Author. 


ELEMENTS  OF  GEOMETRY. 


BOOK  I. 

DEFINITIONS. 

1.  Elementary  Geometry  treats  of  the  properties,  rela- 
tions, and  measurement  of  magnitudes. 

2.  Magnitudes  have  one,  two,  or  three  dimensions ;  as, 
A  line  has  only  one  dimension,  viz.,  length. 

A  surface  has  two,  length  and  breadth. 

And  a  solid  has  three,  length,  breadth,  and  thickness. 

3.  Plane  Geometry  takes  its  name  from  the  plane,  as 
each  figure  is  upon  one  plane. 

Rem. — As  every  figure,  and  every  part  of  it,  is  on  the  same  plane,  it  is 
not  necessary  to  repeat  "  on  the  same  plane." 

4.  A  Mathematical  Plane  is  a  surface  of  indefinite 
extent,  such  that  if  a  straight  edge  or  rule  ba  applied  to  it,  the 
edge  or  rule  will  coincide  with  it,  in  every  position. 

5.  Lines  are  of  two  classes,  straight  and  curved. 

A  Straight  Line  has  everywhere  the  same  direction,  or  it 
may  be  said  to  have  two  directions,  exactly  opposite  each  other, 
from  any  point  in  the  line. 

A  Curved  Line,  or  simply  a  Curve,  constantly  changes 
its  direction. 

6.  Surfaces  are  of  two  classes,  plane  and  curved. 

A  Plane  Surface  corresponds  to  the  mathematical  plane, 
or  a  portion  of  it,  and  it  may  have  any  position  whatever ;  that 
is,  it  may  be  horizontal  or  vertical,  or  it  may  be  oblique. 

A  Curved  Surface  is  such  that  if  a  straight  rule  be 
applied  to  it,  the  rule  will  not  coincide  with  it  in  every  position; 
as  the  surface  of  a  sphere  or  of  a  cylinder. 


b  ELEMENTS     OF     GEOMETKY, 

7.  A  Point  has  position  only ;  as,  any  particular  place  in  a 
line  or  plane,  and  the  extremities  of  lines,  are  called  points. 

8.  A  Circle  is  a  portion  of  a  plane  bounded  by  a  curved 
line,  every  point  of  which  is  equally  distant  from  a  point  within 
called  the  center. 

9.  The  curved  line  is  called  the  Circumference,  and  any 
part  of  it  an  Arc. 

10.  A  Polygon  is  a  portion  of  a  plane  bounded  by  straight 
lines  called  sides. 

A  polygon  of  three  sides 
is  called  a  Triangle. 

A  polygon  of  four  sides 
is  called  a  Quadrilateral.     AZ_  JQ 

A  polygon  of  five  sides  is        quadrilateral. 
called  a  Pentagon. 
A  polygon  of  six  sides  is  called  a  Hexagon,  etc. 

11.  The  divergence  of  any  two  sid^s  from  their  point  of 
intersection  is  called  an  Angle  of  the  polygon  ;  and  the  number 
of  angles  will  always  be  the  same  as  the  number  of  sides  of  the 
polygon. 

J  12.  A  triangle  having  two  equal  sides  is  called  an  Isosceles 

triangle. 

A  triangle  having  three  equal  sides  is  called  an  Equilateral 
triangle. 

A  triangle  having  all  its  sides  unequal  is  called  a  Scalene 
triangle. 

13.  Two  lines  are  Parallel  when  they  are  everywhere 
equally  distant,  and  hence  will  never  meet. 

14.  A  quadrilateral  having  its  opposite  sides  respectively 
parallel  is  called  a  Parallelogram. 

15.  A  quadrilateral  having  only  two  sides  parallel  is  called  a 
Trapezoid. 

16.  A  Regular  Polygon  has  all  its  sides  and  angles 
respectively  equal. 

17.  A  regular  quadrilateral  is  termed  a  Square. 

18.  A  circle,  a  polygon,  etc.,  are  termed  Geometrical 
Figures. 


BOOK     I.  7 

19.  An  Angle  may  be  designated  by  the  letter  at  its  ver- 
tex, or  by  three  letters,  the  letter  at  the  vertex  occupying  the 
middle  place,  and  the  letters  at  the  extremities  of  its  sides  hold- 
ing the  first  and  last  places';  thus,  the  angle  A  in  the  triangle 
ABC  is  designated  angle  BAC. 

20.  When  the  angles  of  a  quadrilateral  are  right  angles,  and 
the  opposite  sides  respectively  equal,  it  is  termed  a  Rectangle. 

21.  The  circumference  of  a  circle  is  divided  into  360  equal 
parts,  called  Degrees,  and  if  radii  be  drawn  to  each  point 
marking  the  degrees,  there  will  be  360  angles,  each  of  one  degree. 


GEOMETRICAL     TERMS. 

1.  An  Axiom  is  a  self-evident  truth. 

2.  A  Theorem  is  a  truth  which  requires  a  demonstration. 

3.  A  Problem  is  a  question  which  requires  a  solution. 

4.  Axioms,  Theorems,  and  Problems  are  Propositions. 

5.  A  Corollary  is  an  obvious  consequence  of  one  or  more 
propositions,  or  of  a  definition. 

6.  A  Scholium  is  a  remark  upon  something  which  pre- 
cedes. 

GENERAL    AXIOMS. 

1.  Magnitudes  which  are  equal  to  the  same  magnitude  are 
equal  to  each  other. 

2.  If  equals  be  added  to  equals,  the  sums  will  be  equal. 

3.  If  equals  be  subtracted  from  equals,  the  remainders  will 
be  equal. 

4.  If  equals  be  added  to  unequals,  the  sums  will  be  unequal. 

5.  If  equals  be  subtracted  from   unequals,  the  remainders 
will  be  unequal. 

6.  If  equals  be  multiplied  by  equals,  the  products  will  be 
equal. 

7.  If  equals  be  divided  by  equals,  the  quotients  will  be  equal. 

8.  The  whole  is  greater  than  any  of  its  parts. 

9.  The  whole  is  equal  to  the  sum  of  all  its  parts. 
10.  Like  powers  and  like  roots  of  equals  are  equal. 


8  ELEMENTS     OF     GEOMETRY. 

SPECIAL    AXIOMS. 

1.  A  straight  line  is  the  shortest  distance  between  two  points. 

2.  Between  two  points  only  one  straight  line  can  be  drawn. 

3.  Two  fixed  points  through  which  a  line  passes  determine 
its  direction. 

Cor.  1.— Two  straight  lines,  having  two  points  common,  form 
one  and  the  same  straight  line. 

Cor.  2. — Two  straight  lines  intersect  at  but  one  point. 

4.  Twro  straight  lines,  starting  from  the   same  point  and 
taking  the  same  direction,  form  one  and  the  same  straight  line. 

5.  Two  straight  lines,  starting  from   the   same   point  and 
taking  different  directions,  form  an  angle. 

6.  Two  straight  lines   starting  from   different  points   and 
taking  the  same  direction,  are  parallel. 

7.  If  two  lines  are  each  parallel  to  a  third,  they  will  be 
parallel  to  each  other. 

8.  Only  one  perpendicular  can  be  drawn  to  a  straight  line, 
either  from  a  point  without  the  line,  or  from  a  point  on  the  line. 


Describe  a  circle,  and  show  the  relation  of  its  -properties, 
and  also  that  of  a  straight  line  touching  it  at  one  point. 

Take  a  string  of  any  definite  length,  say  six  inches,  attach  a 
pin  to  one  end,  and  a  chalk  point  to  the  other  end.     Fasten  the 
pin  at  any  point  in  a  plane  as  a  center,  and,  with  the  string  at 
full  stretch,  revolve  the  chalk  point  around 
the  center,  until  it  reaches  the  point  from 
which  it  started  ;  thus,  let  CA  represent  the 
string,  C  the  center-pin,  and  A  the  chalk- 
point.     ABDE  is  the  chalk-line  made  by  the 
revolution  of  CA;  every  point  in  the  line 
ABDE  will   be  at   the   distance   CA  from 
the  center;   the  curved  line  ABDE  is  the 
circumference  of  the  circle ;  the  portion  of 
the  plane  enclosed  by  it  is  the  circle ;  and  CA  is  the  radius. 


BOOK     I 


/  Def.  1. — Any  straight  line,  as  AB,  pass- 
ing through  the  center  and  terminating  on 
the  circumference,  is  a  Diameter. 

Cor. — A  diameter  is  twice  the  radius. 

Def.  2. — Any  straight  line  as  DE,  touch- 
ing at  but  one  point  as  F,  is  a  Tangent  to 
the  circumference. 

Rem. — A  circumference   can  be   described  with  a  pair  of 
dividers;    the  distance  between   the  points  is  the  Radius  of 

the  circle. 

Two  radii  drawn  from  the  center  of  a  circle  to  its  circumfer- 
ence, form  an  angle  of  as  many  degrees  as  is 
contained  in  the  arc  intercepted  by  its  sides. 

Def.  1. — When  the  angle  is  90  degrees, 
it  is  called  a  Right  Angle. 

Def.  2. — When  the  angle  is  less  than  90 
degrees,  it  is  called  an  Acute  Angle. 

Def.  3. — When  the  angle  is  greater  than 
90  degrees,  it  is  called  an  Obtuse  Angle. 

Def.  4. — The  Complement  of  an  angle  is  the  difference 
between  the  angle  and  90  degrees. 

Cor. — If  the  sum  of  two  angles  is  90  degrees,  the  one  is  the 
complement  of  the  other. 

Def.  5.— The  Supplement  of  an  angle  is  the  difference 
between  the  angle  and  180  degrees. 

Cor. — If  the  sum  of  two  angles  is  180  degrees,  the  one  is  the 
supplement  of  the  other. 


10 


ELEMENTS     OF     GEOMETRY. 


THEOREM    I. 

Tit e  _  diameter  of  a  circle  bisects  the  circle  and  its  cir- 
cumference. 

Let  AB  be  the  diameter  of  the   circle 
ADBE. 

Revolve  the  part  ADB  upon  AB  as  an  axis, 
until  it  falls  upon  AEB.  The  arc  ADB  will 
coincide  with  the  arc  AEB;  otherwise  some 
points  in  the  circumference  would  be  un- 
equally distant  from  the  center  of  the  circle ; 
hence  the  part  of  the  circle  ADB  is  equal  to 
the  part  AEB;  and  the  arc  ADB  is  equal  to  the  arc  AEB. 

Cor. — Each  one  of  the  two  equal  parts  of  the  circle  is  a  semi- 
circle, and  the  corresponding  arcs  are  semi-circumferences. 


THEOREM    II. 

An  angle  at  the  center  of  the  circle  is  measured  by  the 
arc  intercepted  by  its  sides. 

Let  C  be  the  center  and  AB  the  diameter 

of  a  circle,  A'B'  a  two-pointed  needle,  with  a 

pivot  at  the  center  C  about  which  it  revolves. 

Since  A'B'   passes  through  the   center  and 

terminates  in   the   circumference,  it    is    a 

diameter,  and  in  every  position  bisects  the 

circle  and  its  circumference  -(Theorem  1) ; 

hence,  as  A'  is  moved  towards  E,  B'  moves 

towards  D;  the  arcs  AA'  and  BB'  are  constantly  equal ;  and  the 

radii  CA  and  CA',  also  CB  and  CB',  make  equal  angles  at  the 

center  C. 

When  A'  reaches  E,  90  degrees  from  A;  B'  will  be  at  D, 
90  degrees  from  B  ;  and  there  will  be  four  equal  angles  at  C, 
each  90  degrees  ;  and  the  diameters  are  said  to  be  at  right  angles, 
or  perpendicular  to  each  other. 

As  the  arc  AA'  increases  by  one,  two,  etc.,  degrees,  so  also 
the  angle  ACA'  increases  by  the  same  number  of  degrees. 


BOOK     I. 


11 


THEOKEM    III. 

If  one  straight  line  intersect  another  straight  line,  the 

sum  of  any  two  adjacent  angles  will  be  equal  to  tiro 

right  angles. 

* 

Let  the  two  straight  lines  intersect  at  C, 
then  with  C  as  a  center  and  any  radius 
describe  a  circumference  cutting  the  lines 
at  A,  E,  B  and  D.  Since  AB  is  a  diameter, 
the  two  angles  ACE  and  ECB  will  be  meas- 
ured by  the  sum  of  the  two  arcs  AE  and  EB, 
which  is  equal  to  a  semi-circumference  or 
two  right  angles.  So  also  the  two  angles  ACD  and  BCD,  and 
since  DE  is  a  diameter,  the  sum  of  ACD  and  ACE  is  equal  to  two 
right  angles,  also  the  sum  of  ECB  and  BCD. 

Cor.  1. — Vertical  angles  are  equal,  as  each  one  is  the  supple- 
ment of  the  same  angle;  thus,  ACD  and  ECD  is  each  the  supple- 
ment of  ACE,  or  its  equal  BCD. 

Cor.  2. — The  sum  of  all  the  angles  at  a  point  on  each  side  of  a 
straight  line  is  equal  to  two  right  angles  ;  and  the  sum  of  all  the 
angles  around  a  point  is  equal  to  four  right  angles. 

Cor.  3. — Equal  arcs  have  equal  radii  and  are  like  parts  of 
equal  circumferences. 

Cor.  4. — Equal  angles  have  equal  arcs,  and  equal  arcs  have 
equal  chords,  the  radii  being  equal. 

Scho. — If  several  circumferences,  with  different  radii,  be  de- 
scribed from  the  same  center,  the  circumferences  will  be  parallel. 


THEI1EM    IV. 

TJie  diameter  of  a  circle  is  greater  than  any  other  chord. 

Let  AB  be  the  diameter  and  B#  a  chord 
of  a  circle.  Draw  the  radius  CP,  which  is 
equal  to  CA;  6fr  is  less  than  the  sum  of 
BC    and    Ct>,    (Special    Axiom    1);     but 

BC  +  CD  =  AC  +CB  =  AB  ;  therefore  DB 
is  less  than  AB,  or  AB  >  DB. 


12 


ELEMENTS     OF     GEOMETRY. 


THEOREM    V. 

^j^-J"/  two  straight  lines  meet  a  third  line,  making  any  two 
angles  which  are  similarly  situated  with  regard,  to  the  two 
lines,  and  on  the  same  side  of  the  third  line  equal,  then 
will  the  two  lines  be  parallel.     The  converse  is  also  true. 

If  the  two  lines  CD  and  EF  meet  AB, 
laking  the  angles  AGD   and   AHF   equal, 
^hen  will  CD  and  EF  be  parallel. 

CD  and  EF  may  be  regarded  as  starting 
different  points  G  and   H,  and  as  they 
pmake  the  angles  AGD  and  AHF  equal,  they 
take  the  same  direction  and  are  therefore 
parallel.     (Special  Axiom  6.) 
The  converse  is  necessarily  true. 

Cor.  1. — The  same  is  true,  when  the  equal 
angles  are  right  angles  ;  hence,  two  lines  per- 
pendicular to  a  third  are  parallel. 

Cor.  2. — Since  the  angles  marked  i  and  i 
are  equal,  and  their  vertical  angles  are  also 
equal,  hence  four  of  these  angles  are  equal; 
and  as  each  of  the  remaining  four  is  supple- 
mentary to  one  of  these,  they  are  consequently  equal. 

Cor.  3. — If  one  of  these  angles  is  acute,  four  will  be  acute, 
and  the  other  four  will  be  obtuse  ;    but  if 
one  is  a  right  angle,  all  will  be  right  angles. 
Def.  1. — Angles   similarly    situated  are 
called  corresponding  angles. 

Cor.  1. — If  two  parallels  are  cut  by  a  third 
line,  the  corresponding  angles  will  be  equal. 

Cor.  2. — The  interior  angles  on  the  same  side  are  supplementary. 


C— ^ 


^/ 

2/1 


Cor.  2.  Cor.  8.  Cor.  4. 

Cor.  3. — The  alternate  exterior  angles  are  equal. 
Cor.  4. — The  alternate  interior  angles  are  equal. 
Schc. — In  the  above  figures  the  same  numbers  indicate  pairs. 


BOOK     I. 


13 


THEOREM    VI. 

Two  angles,  having  their  sides  parallel  and  lying  in  the 
same  or  in  opposite  directions,  are  equal. 

Let  AB  and  DE  be  parallel,  also  BC 
and  EF,  and  lying  in  the  same  direction  ; 
then  will  the  angles  ABC  and  DEF  be 
equal. 

Produce  DE  to  H,  cutting  BC  in  G. 
Since  the  parallels  are  cut  by  DH,  the 
corresponding  angles  DEF  and  DGC 
are  equal ;  and  as  BC  cuts  the  parallels  DH  and  AB,  the 
angles  DGC  aj^d  ABC  are  corresponding  angles  and  hence  are 
equal ;  therefore,  the  angle  ABC  is  equal  to  the  angle  DEF. 
(Ax.  1.) 

The  angles  DGC  and  BGH  are  vertical  angles,  therefore  equal. 
Consequently  the  angles  ABC  and  BGH  are  equal. 


x 


THEOREM    VII. 


Two  angles,  having  their  sides  'respectively  perpendicu- 
lar, are  equal  or  supplementary. 

\  <•  Let  the  sides  of  the  angle  EAD  be  respec- 
*  fively  perpendicular  to  the  sides  of  the  angle 
BAC;  and  also  the  sides  of  EAD'  perpendicular 
to  the  sides  of  BAC. 


1st.  The  angles  BAD  and  CAE  are  right 
angles ;  from  each  take  the  angle  CAD,  and 
there  remains  the  angle    BAC   equal    to   the  d' 

angle  DAE. 

2d.  The  angle  EAD'  is  supplementary  to  EAD  ;  so  also  of  its 
equal  BAC. 


14 


ELEMENTS     OF     GEOMETEY, 


THEOREM    VIII. 

If,  in  a  circle,  two  diameters  be  drawn  at  right  angles, 
and  several  chords  be  drawn  -parallel  to  one  of  the  diame- 
ters, and  at  the  extremities  of  the  other  diameter,  lines  be 
drawn  parallel  to  the  chords, 

1st.  Tlie  chords  will  be  bisected  by  the  perpendicular 
diameter. 


2d.   The  lines  at  the  extremities  of  the  same  diameter, 
will  be  tangents  to  the  circumference. 

3d.  Any  two  parallels  will  intercept  equal  arcs  of  the 

circumference. 

» 

Since  DE  is  a  diameter,  DAE  is  a  semi- 
circle, and  if  it  be  revolved  upon  DE  as  an 
axis,  until  it  fall  upon  DBE,  the  two  semi- 
circles will  coincide;  and  since  all  the  angles 
made  with  DE  by  the  diameter  AB  and  each 
line  parallel  to  AB  are  right  angles,  all  the 
parts  of  the  lines  of  the  semicircle    DAE 
will    fall  upon    and   coincide    with    those 
of   the    other    semicircle;     that    is,    CA   with    CB;    FL  with 
LG;  and  HK   with  Kl  ;    also   MD  with  DN  and  PE  with  EQ; 
therefore  ; 


M —^ 

r/ 

L          \« 

A  ( 

IB 

P        — ^»- 

C                J 

.  ^   Q 

1st.  The  chords  are  bisected. 

2d.  The  two  lines  MN  and  PQ  can  only  touch  the  circumfer- 
ence at  D  and  E  respectively ;  for,  at  these  points,  the  straight 
lines  and  the  curves  may  be  regarded  as  starting,  and  taking 
different  directions,  for  the  straight  lines  are  parallel  to  the 
chords,  whilst  the  curves  approach  and  intersect  them. 

3d.  As  the  one  half  of  each  chord  falls  upon  its  other  half, 
and  the  one  side  of  each  tangent  falls  upon  its  other  side,  so  also 
the  intercepted  arcs  respectively  fall  upon  and  coincide  with  each 
other,  and  hence  are  equal. 

Cor.  1. — A  radius  perpendicular  to  a  chord  bisects  the  chord 
and  also  its  arc. 


BOOK     I.  15 


■    Co 


)ok.  2. — A  tangent   is  perpendicular  to    a   radius  at  its 
extremity. 

Cor.  3. — A  line  perpendicular  to  a  chord  at  its  middle  point, 
passes  through  the  center  of  the  circle. 

Scho. — Observe  that  a  tangent  touches  the  circumference  at 
but  one  point,  and  a  chord  intersects  the  circumference  at  two 
points,  each  end  passing  away  in  opposite  directions  ;  hence  a 
straight  line  can  only  intersect  a  circumference  at  two  points. 


THEOREM    IX. 

If  a  perpendicular  be  erected  at  the  middle  point  of  a 
straight  line,  every  point  in  the  perpendicular  is  equally 
distant  from  the  extremities  of  the  line. 

Let  PC  be  perpendicular  to  AB  at  its 
middle  point  C;  then  will  any  point  in 
the  line  PC  be  equally  distant  from  A 
and  B. 

Take  any  point  in  the  perpendicular 
PC  as  D,  and  draw  AD  and  BD  ;  and  let 
the  part  ACD  be  revolved  on  DC  as  an  axis,  until  it  fall  upon 
the  plane  of  BCD;  since  both  angles  at  C  are  right  angles,  CA 
will  take  the  direction  of  CB,  and  as  AC  is  equal  to  CB,  the 
point  A  will  fall  upon  the  point  B,  and  CA  will  coincide 
with  CB,  and  AD  must  fall  upon  and  coincide  with  DB. 
(Special  Ax.  2.} 

Cor.  1. — As  two  points  determine  the  direction  of  aline;  any 
straight  line  which  has  two  points  equally  distant  from  the 
extremities  of  another  line,  is  perpendicular  to  the  latter  at  its 
middle  point. 

Cor.  2. — With  D  as  a  center  and  DA  as  a  radius,  a  circum 
ference  may  be  described  which  will  pass  through  the  points  A 
and  B,  and  AB  becomes  a  chord  of  the  circumference ;  and  as  a 
straight  line  cannot  intersect  a  circumference  at  more  than  two 
points,  there  can  be  only  two  points  in  the  line  AB  equally 
distant  from  the  point  D. 


x 


16  ELEMENTS     OF     GEOMETRY. 


> 


THEOREM    X 


A  -perpendicular  is  the  shortest  distance  from,  the 
center  of  a  circle  to  a  chord,  or  from  a  point  to  a  line. 

Let  AB  and  EF  be  two  parallel  chords, 
and  draw  CD  perpendicular  to  AB ;  it  will 
also  be  perpendicular  to  EF.  At  the  point  D, 
the  extremity  of  the  radius  CD,  draw  GH 
perpendicular  to  CD,  and  it  will  be  parallel 
to  the  chords  AB  and  EF. 

1st.  The  perpendicular  CK  is  less  than  _ 
CA  or   CB,   each  of  which  is  equal  to  CD,,  of  which  CK  is 
a  part. 

CI  is  less  than  CE  or  CF  for  the  same  reason. 

As  the  chord  departs  from  the  center  and  consequently 
diminishes,  the  perpendicular  approaches  the  radius  in  length, 
but  can  never  equal  it  whilst  the  chord  has  any  definite 
length. 

2d.  CD  is  less  than  any  oblique  line  drawn  from  the  point 
C  to  GH;  for  any  oblique  line  as  CL  will  terminate  without 
the  circumference,  and  consequently  be  greater  than  the 
radius. 

Cor.  1. — A  perpendicular  is  the  shortest  distance  from  a 
point  to  a  line,  and  also  between  two  parallels. 

Cor.  2. — The  farther  distant  from  the  center,  the  less  the 
chord. 

Cor.  3. — The  less  the  chord  the  less  the  arc,  and  conse- 
quently the  less  the  opposite  angle. 

is ' 


BOOK     I. 


17 


r  PROBLEM    I. 

To  bisect  a  given  line. 

Let  AB  be  the  given  line ;  then  with  A  and 
B  as  centers,  and  a  radius  greater  than  the 
half  of  AB,  describe  arcs  above  and  below  the 
line  AB,  intersecting  at  D'  and  E,  and  join 

D'  and  E,  cutting  AB  in  C,  which  will  be  the     Al 

middle  point.     (Th.  9,  Cor.  1.) 

Sch.  1. — The  intersections  may  both  be 
made  on  the  same  side  of  AB,  as  at  D'  and  D, 
by  taking  different  radii. 

Sch.  2. — As  the  radius  becomes,  as  it  were,  an  oblique  line, 
whilst  one-half  of  AB  is  a  perpendicular,  it  must,  of  course,  be 
greater  than  one-half  of  AB. 


PROBLEM    II. 

From  a  point  without  a  line,  to  draw  a  perpendicular 
to  the  line. 

Let  P  be  a  point  without  the  line  CD. 
With  P  as  a  center,  and  a  radius  greater 
than  the  shortest  distance  to  CD,  which 
would  be  a  perpendicular,  draw  an  arc  cut- 
ting CD  in  A  and  B;  then  with  A  and  B  as 
centers  and  a  radius  greater  than  one-half  of 
AB,  describe  arcs  intersecting  at  E  ;  then 
will  P  and  E  be  two  points  equally  distant  from  A  and  B,  and 
hence  PE  is  perpendicular  to  AB  or  to  CD.     (Th.  9,  Cor.  1.) 


BVP 


PROBLEM»III. 
At  a  point  in  a  line,  to  erect  a  perpendicular  to  the  line. 

Let  P  be  a  point  in  the  line  CD;  then, 
with  P  as  a  center  and  a  radius  PA,  cut  CD 
in  two  points,  A  and  B ;   and  with  A  and  B 

respectively  as  centers,  and  a  radius  greater       ^-\ 

than  one-half  of  AB,  describe  arcs  intersect- 
ing at  E,  and  join  PE.     It  will  be  perpendicular  to  CD  at  the 
point  P. 

X 


18 


ELEMENTS     OF     GEOMETRY 


PROBLEM    IV. 

To  bisect  a  given  angle. 

Let  BAC  be  the  given  angle.  Then 
with  A  as  a  center  and  a  radius  that  will 
cut  the  sides  AB  and*  AC, .draw  the  arc  DE 
and  its  chord  ;  then  with  D  and  E,  respec- 
tively, as  centers  and  a  radius*  greater  than 
the  half  of  DE  draw  arcs  intersecting  at  F,  and  join  AF.  The 
two  points  A  and  F  will  be  equally  distant  from  D  and  E ;  hence 
the  line  AF  will  bisect  the  chord  DE,  its  arc,  and  hence  the 
angle  A. 


\      .  PROBLEM    V. 

From  a  point  without  a  line,  to  draw  a  parallel  to  the 

line. 


-k 


Let  P  be  a  point  without  the  line 
CB.  'From  P  draw  PA  perpendicular 
to  CB  (Prob.  2),  and  from  P  draw  a 
perpendicular  to  AP.  Then  will  PQ 
be  parallel  to  CB.     (Th.  5.) 


PROBLEM    VI. 


\  \  To  find  the  center  of  a  given  circle. 

Draw  any  two  chords,  as  AB  and  BC,  to 
the  gjven  circle,  and  pass  perpendiculars 
through  their  middle  points;  both  perpen- 
diculars will  pass  through  the  center  of  the 
circle.  (Th.  8,  Cor.  3.)  The  point  of  their 
intersection  0,  which  is  the  only  common 
point,  is  the  center  of  the  circle. 

Scho. — It  is  not  necessary  that  the  chords  be  consecutive,  but 
they  must  not  be  parallel,  as  then  there  would  be  but  one  per- 
pendicular and  the  same  perpendicular  would  pass  through  the 
middle  points  of  both  chords. 


BOOK    I. 


'19 


^  f  PROBLEM    VII. 

To  circumscribe  a  circle  about  a  triangle. 

Let  ABC  be  the  given  triangle.  Pass 
the  perpendiculars  DO  and  EO  through 
the  middle  points  of  any  two  sides  of  the 
triangle,  as  AC  and  CB;  their  point  of 
intersection  0  will  be  the  center  of  a  circle 
of  which  AC  and  CB  are  chords. 

Cor.  1. — A  circumference  can  be  passed 
through  any  three  points  not  in  the  same  straight  line;  but  if 
the  three  points  are  in  the  same  straight  line,  only  one  perpen- 
dicular can  be  drawn,  and  hence  no  solution. 

Cor.  2. — Each  of  the  three  points  is  equally  distant  from  the 
center,  but  only  two  points  in  the  same  straight  line  can  be 
equally  distant  from  a  point  without  the  line.     (Th.  9,  Cor.  2.) 


BOOK     II 


Def. — An  Inscribed  Angle  has  its  vertex  in  the  circum- 
ference of  a  circle  of  which  its  sides  are  chords. 

THEOREM    I. 

An  inscribed  angle  is  measured  by  one-half  the  arc 
intercepted  by  its  sides. 


FIRST  CASK. 


SECOND  CASE. 


THIRD  CASE. 


There  are  three  cases  : 

1st.  When  one  of  its  sides  is  a  diameter;  as,  the  angle  BAD 
has  one  side  AB  a  diameter.  Through  the  center  C  draw  EF 
parallel  to  the  chord  AD ;  then  will  the  angles  BAD  and  BCF 
be  equal,  as  they  are  corresponding  angles ;  but  BCF  and  ACE 
are  equal,  as  they  are  vertical  angles — they  are  both  angles  at 
the  center,  measured  respectively  by  the  arcs  FB  and  AE,  which 
arcs  are  consequently  equal.  Therefore  arc  AE  is  equal  to  arc 
FB,  also  equal  to  arc  DF;  hence  FB  is  one-half  of  BD.  There- 
fore the  angle  A  is  measured  by  £  arc  BD,  which  is  the  arc  inter- 
cepted by  its  sides. 

2d.  When  the  center  of  the  circle  is  without  the  angle,  as  the 
angle  BAD. 

Here  angle  EAD  is  measured  by  £  arc  DE, 

And  "      BAE  «        "  "  \    "    BE. 

By  subtraction,     "      BAD  "       *  "  £  "    BD. 

3d.  When  the  center  is  within  the  triangle ;  as  BAD. 
Here  angle  BAE  is  measured  by  J  arc  BE, 

And  "      DAE  "        "  "  \  "    DE. 

By  addition,  "      BAD"        "  "i"    BD. 


V- 


I    u 


BOOK     II 


21 


THEOREM   I.  — Continued 


Cor.  1. — All  the  angles  inscribed  in  the 
same  segment  are  equal;  since  the  angles 
A,  B,  and  C  are  measured  each  by  the  half 
of  the  same  arc  DwE,  they  are  equal. 


Cor.  2. — An  angle  inscribed  in  a  semicircle  is  a  right  angle, 
as  BAD. 

I 

Cor.  3. — An  angle  inscribed  in  a  segment  greater  than   a 

semicircle  is  acute,  as  BAC ;  and  an  angle  inscribed  in  a  segment 
less  than  a  semicircle  is  obtuse,  as  BDC. 


Cor.  2. 


Cor.  4. 


Cor.  4. — The  opposite  angles  of  an  inscribed  quadrilateral 
are  supplementary;  as, 

Angle  A  measured  by  |  arc  BCD. 
"       C  "         "   "    "    DAB. 

The  sum  of  the  two  arcs  is  the  circumference ;    hence    half 
their  sum  is  the  measurement  of  two  right  angles. 


Cor.  5. — If  the  extremities  of  the  chords 
forming  an  inscribed  angle  be  joined  by  a 
straight  line,  an  inscribed  triangle  is  formed. 
As  in  the  angle  A  join  B  and  C ;  then  the 
triangle  ABC  has  each  of  its  angles  inscribed, 
and  is  therefore  an  inscribed  triangle. 


22  ELEMENTS     OF     GEOMETRY 


THEOREM    II. 

( 
TJie  sum  of  the  three  angles  of  any  triangle  is  equal  to 
two  right  angles. 

Let  ABC  be  an  inscribed  triangle  ;  then  will 

angles  A  +  B  +  C  equal  two  right  angles ;  as, 

The  angle  A  is  measured  by  J  arc  BC, 
u       u    g  u       u         n  «   «  %Qf 

a  a      Q  n  a  a    a     a     ad 

The  sum  of  the  three  arcs  is  a  circumference, 
one-half  of  which  measures  the  angles  and  is 
the  measure  of  two  right  angles ;  hence,  the 
sum  of  the  angles  of  an  inscribed  triangle  is  equal  to  two  right 
angles;  but,  as  a  circumference  may  be  passed  through  all  the 
vertices  of  any  triangle  and  the  triangle  become  inscribed, 
(Book  I,  Problem  7),  it  follows  that  the  sum  of  the  three  angles 
of  any  triangle  is  two  right  angles. 

Cor.  1. — If  the  triangle  is  isosceles,  'two  of  its  angles  will  be 
equal. 

Cor.  2. — If  the  triangle  is  equilateral,  all  the  angles  will  be 
equal,  each  60  degrees. 

Cor.  3. — If  the  triangle  is  scalene,  all  the  angles  will  be 
unequal. 

Cor.  4. — As  an  inscribed  angle  is  measured  by  half  the  arc 
intercepted  by  its  sides,  and  the  greater  the  arc  the  greater  the 
chord,  hence  the  greatest  angle  is  opposite  the  greatest  chordWid 
the  next  to  the  greatest  angle  opposite  the  chord  next  to  the 
longest,  and  the  smallest  angle  opposite  the  shortest  chord ;  con- 
sequently, in  any  triangle  the  greatest  angle  is  opposite  the  longest 
side,  the  next  to  the  greatest  angle  opposite  the  next  to  the  longest 
side ;  and  the  smallest  angle  opposite  the  shortest  side. 

Scho.  1. — If  a  triangle  has  two  sides  and  the  included  angle 
given,  the  three  vertices  of  the  triangle  are  fixed,  and  the  triangle 
determined. 

Scho.  2. — One  side  and  the  two  adjacent  angles  fix  the  three 
vertices,  but  if  one  of  the  given  angles  be  the  opposite  angle,  the 
other  adjacent  angle  is  the  supplement  of  the  sum  of  the  two 
given  angles. 

Scho.  3. — The  three  sides  of  a  triangle  also  determine  the 
triangle. 


BOOK     II 


;>:; 


Coe. — Two  triangles,  each  having  the  three  parts  named  in 
either  of  the  scholia  respectively  equal,  are  equal  in  all  their  parts. 

Rem. — The  sum  of  the  angles  of  a  triangle  may  be  determined 
by  means  of  the  parallels,  as  follows  : 

Let  ABC  be  any  triangle.  At  C  draw  DE 
parallel  to  AB.  The  angles  marked  1  and  1 
and  2  and  2  are  respectively  alternate  inte- 
rior angles,  and  consequently  respectively 
equal ;  hence,  the  three  angles  of  the  triangle  are  equal  to  all  the 
angles  at  a  point  on  one  side  of  a  straight  line,  which  is  two 
right  angles.     (Book  I,  Th.  3,  Cor.  2.) 


THEOREM    III 


If  one  side  of  a  triangle  is  produced  in  one  direction, 
the  exterior  angle  formed  is  equal  to  the  sum  of  the  two 
interior  angles  not  adjacent. 

Let  ABC  be  a  triangle.  Produce  BC 
to  D,  forming  the  exterior  angle  ACD. 
From  C  draw  CE  parallel  to  BA ;  then 
will  the  angles  marked  1  and  1  be  cor- 
responding angles   and  equal,  and   the 

angles  marked  2  and  2  be  alternate' interior  angles  and  equal: 
hence,  the  exterior  angle  ACD  is  equal-  to  the  sum  of  ABC  and 
BAC,  the  two  interior  angles  not  adjacent. 


THEOREM    IV 


Every  -point  in  the  line  which  bisects  an  angle  is  equally 
distant  from  each  side  of  the  angle. 

Let  AP  bisect  the  angle  A,  and  revolve 
the  part  CAP  on  AP  as  an  axis  ;  AC  will  fall 
upon  and  coincide  with  AB,  since  angle  CAP 
is  equal  to  angle  PAD.  From  any  point  as 
P  draw  PD  perpendicular  to  AB  ;  it  will  be 
the  shortest  distance  to  AB,  and  also  to  AC,  which  coincides 
with  AB. 


s. 


24:  ELEMENTS     OF     GEOMETRY 


THEOREM    V. 

If  from  a  point  without  a  line  a  perpendicular  be 
drawn  to  the  line,  and  oblique  lines  to  different  points  of 
the  line  : 

1st.  The  perpendicular  will  be  shorter  than  any  oblique 
line. 

2d.  Any  two  oblique  lines  at  equal  distances  from  the 
foot  of  the  perpendicular  will  be  equal. 

Sd.  TJie  farther  from  the  foot  of  the  perpendicular,  the 
greater  the  oblique  line. 

1st.  In  the  triangle  ABD,  the  angle  B  is  a 
right  angle ;  hence  it  is  greater  than  the  angle 
D.  Therefore  the  side  AB  opposite  the  angle 
D,  is  less  than  the  side  AD  opposite  the  angle  B. 

2d.  The  triangles  ABD  and  ABC  have  each 
two  sides  and  the  included  angle  respectively 
equal ;  hence  the  triangles  are  equal,  and  AC  equal  to  AD. 

3d.  In  the  triangle  ACE,  the  angle  ACE  is  obtuse,  conse- 
quently greater  than  the  angle  AEC;  therefore  the  side  AE  is 
greater  than  the  side  AC. 


THEOREM    VI. 

The'  sum  of  all  the  angles  of  any  quadrilateral  is  equal 
to  four  right  angles. 

Let  A  BCD  be  any  quadrilateral.  Draw 
the  diagonal  DB,  dividing  the  quadrilateral 
into  two  triangles.  All  the  angles  of  the 
two  triangles  make  up  precisely  the  angles 
of  the  quadrilateral ;  but,  the  sum  of  all  the 
angles  of  the  two  triangles  is  four  right  angles.  Hence,  the  sum 
of  all  the  angles  of  any  quadrilateral  is  four  right  angles. 


BOOK     II, 


25 


0 


THEOREM    VII 


The  sum  of  all  the  angles  of  any  -polygon  is  equal  to 
two  right  angles  taken  as  many  times  as  the  -polygon  has 
sides,  minus  four  right  angles. 

Let  ABCDE  be  any  polygon.  Take 
any  point  within  the  polygon,  as  0,  and 
from  it  draw  lines  to  the  extremities  of 
all  the  sides.  The  number  of  triangles 
will  be  equal  to  the  number  of  sides  of 
the  polygon.  The  sum  of  the  angles  of 
each  triangle  is  two  right  angles ;  hence, 

the  sum  of  all  the  angles  of  the  triangles  which  make  up  the 
polygon,  is  two  right' angles  taken  as  many  times  as  the  polygon 
has  sides;  but  all  the  angles  at  0,  which  equal  four  right  angles, 
belong  to  the  triangles,  but  not  to  the  polygon,  and  must  be 
deducted  from  the  sum  of  all  the  angles  of  the  triangles,  and  the 
difference  will  be  the  angles  of  the  polygon;  therefore,  the  sum 
of  all  the  angles  of  any  polygon  is  equal  to  two  right  angles 
taken  as  many  times  as  the  polygon  has  sides  minus  four  right 
angles. 


THEOREM    VIII. 


If  each  side  of  a  polygon  is  prolonged,  the  sum  of  all 
the  exterior  angles  thus  formed  will  be  equal  to  four  right 
angles* 

At  each  vertex  of  the  polygon,  the  sum 
of  the  interior  and  exterior  angles  is  two 
right  angles ;  hence,  the  sum  of  all  the  in- 
terior and  exterior  angles  is  equal  to  two 
right  angles  taken  as  many  times  as  the 
polygon  has  sides,  which  sum  is  four  right 
angles  more  than  the  sum  of  all  the  interior  angles.  (Th.  8.) 
Therefore,  the  sum  of  all  the  exterior  angles  is  four  right 
angles. 

2 


X 


G) 


ELEMENTS     OF     GEOMETRY. 


THEOREM    IX 


The  opposite  sides  and  opposite  angles  of  a  parallelo- 
gram are  respectively  equal. 

Let  A  BC  D  be  a  parallelogram.  Draw 
the  diagonal  DB.  Since  AB  and  DC 
are  parallels  cut  by  DB,  the  angles  1  and 
1  are  alternate  and  equal;  and  since  AD 
and  BC  are  parallels  cut  by  DB,  the 
angles  2  and  2  are  alternate  and  equal.  The  triangles  ABD  and 
BCD  have  the  side  BD  common  and  the  adjacent  angles  equal; 
hence  the  triangles  are  equal.  Therefore,  AB  is  equal  to  DC,  and 
AD  equal  to  BC.  The  angles  3  and  3  are  equal,  and  the  sums  of 
the  same  angles  at  B  and  D  are  equal. 


THEOREM    X. 


If  the  opposite  sides  of  a  quadrilateral  are  respectively 
equal,  it  will  be  a  parallelogram* 

Let  AB  equal  DC,  and  AD  equal  BC. 
Draw  the  diagonal  DB.  The  triangles  ABD 
and  BCD  have  their  three  sides  respectively 
equal;  hence  the  triangles  are  equal,  and 
the  angles  opposite  the  equal  sides  equal,  that  is,  1  equals  1,  and 
2  equals  2,  and  these  are  respectively  alternate  angles;  hence,  the 
opposite  sides  are  parallel,  and  the  quadrilateral  is  a  parallelogram. 


THEOREM    XI. 

If  two  opposite  sides  of  a  quadrilateral  are  equal  and 
parallel,  the  figure  will  be  a  parallelogram. 

Let  AB  and  DC  be  equal  aud  paral- 
lel, and  draw  the  diagonal  DB. 

Since  AB  and  DC  are  parallel,  the 
alternate  angles  1  and  1  are  equal,  and 
the  triangles  ABD  and  DCB  have  re- 
spectively two  sides  and  an  included  angle  equal,  and  are  there- 
fore equal,  which  makes  the  other  sides  equal  and  parallel,  ami 
the  opposite  angles  of  the  figure  equal;  hence  it  is  a  par- 
allelogram. 


0 


BOOK     II, 


THEOREM    XII 


27 


The   diagonals  of  a  -parallelogram    mutually    bisect 
each  other. 

The  triangles  ABE  and  DCE  have  a 
side  and  two  adjacent  angles  respec- 
tively equal ;  hence  the  side  AE  oppo- 
site the  angle  2,  is  equal  to  EC 
opposite  the  angle  2  in  the  other 
triangle,  and  DE  is  equal  to  EB  for  the  same  reason. 


THEOREM    XIII.* 

An  angle  formed  by  a  tangent  and  a  chord  is  measured 
by  one- half  the  intercepted  arc. 

The  angle  BAD,  formed  by  the  tangent 
AD  and  the  chord  AB,  is  measured  by  J- 
arc  Aw?B. 

From  B  draw  the  chord  BE  parallel 
to  the  tangent  AD,  then  the  angles  BAD 
and  ABE  are  alternate  interior  angles  and 
consequently  equal.  The  angle  ABE  is 
inscribed,  and  is  measured  by  £  arc  A?iE, 
which  is  equal  to  the  arc  AmB,  as  they  are  intercejrod  by 
two  parallels;  consequently  the  angle  BAE  is  measured  by  \ 
arc  kmB. 

THEOREM    XIV. 

An  angle  formed  by  two  chords  intersecting  ivithin  the 
circle,  is  measured  by  one-half  the  sum  of  the  intercepted 
arcs. 

Let  AB  and  DE  be  two  chords  intersect- 
ing at  C.  ■  ^ 

From  A  draw  the  flV  AF  parallel  to 
DE,  then  will  the  anglel^AF  and  BCE  be 
corresponding  angles,  and  equal ;  the  angle 
BAF  is  inscribed,  and  is  measured  by  J-  arc 
BEF;   but  the  arc  FE  is  equal  to  the  arc  AD,  x 

(therefore  arc   BEF  is  equal  to  the  sum  of  the  arcs  EB  and  AD  ;  j 
consequently  the  angle  BCE,  or  its  equal  ACD,  is  measured  by  J 
the  sum  of  the  arcs  included  by  its  sides. 


S 


v; 


28 


ELEMENTS     OF     GEOMETRY 


THEOREM    XV. 

An  angle  formed  by  two  secants  meeting  without  th4 
circle,  is  measured  by  one-half  the  difference  of  the  inter- 
cepted  arcs. 

The  angle  A  is  formed  by  the  two 
secants  AB  and  AD. 

From  C  draw  CE  parallel  to  AD; 
BAD  and  BCE  are  corresponding 
angles.  The  angle  BCE  is  measured 
by  \  arc  BE  =  BD  —  CF;  therefore, 
the  angle  A,  formed  by  two  secants  meeting  without  the  circle,  is 
measured  by  one-half  the  difference  of  the  intercepted  arcs.. 


THEOREM    XVI. 

An  angle  formed  by  a  tangent  and  a  secant  meeting 
without  a  circle,  is  measured  by  one-half  the  difference  of 
the  intercepted  arcs. 

The  angle  BAD  is  formed,  by  a 
tangent  and  a  secant  meeting  at  A. 
From  C  draw  CE  parallel  to  AB; 
then  the  angles  BAD  and  DCE  are 
corresponding  and  equal ;  but  the 
angle  DCE  is  inscribed,  and  meas- 
ured by  \  arc  DE,  and  DE  = 
DB  —  BE,  or  its  equal  BC;  parallels  intercept  equal  arcs,  there- 
fore the  angle  A  is  measured  by  £  arc  DE  =  £  arc  (DB  —  €C). 


THEOREM    XVII. 

An  angle  formed  by  two  tangents  meeting  without  a 
circle,  is  measured  by  one-half  the  difference  of  the  inter- 
cepted arcs. 

The  angle  A  is  formed  by  two 
tangents  meeting  without  the  cir- 
cle. At  C  draw  CD  parallel  to  AB, 
then  the  angles  BAC  and  DCE  will 
be  corresponding  and  equal ;  but  the 
angle  DCE  is  formed  by  a  tangent 
and  a  chord,  and  hence  is  measured 


BOOK     II 


29 


by  4  arc  CmD.  (Th.  14.)  Arc  DmC  =  (arc  BrDmC  —  arc 
BrD)  and  arc  BrD  =  arc  BnC  ;  therefore,  an  angle  formed  by 
two  tangents  intersecting  without  a  circle  is  measured  by  one- 
half  the  difference  of  the  intercepted  arcs. 

THEOEEM    XVIII. 

The  side  of  a  regular  hexagon  is  equal  to  the  radius 
of  the  circumscribed  circle. 

Describe  a  circumference  and  make  the 
chord  AB  equal  to  the  radius  of  the  circle. 
Draw  the  radii  CA  and  CB.  CAB  will  be 
an  equilateral  triangle,  each  side  equal  to  the 
radius  of  the  circle,  and  each  angle  equal  to 
60  degrees  ;  hence,  AB  is  a  chord  of  an  arc  of 
sixty  degrees,  which  is  contained  exactly  six 
times  in  the  circumference,  and  is  therefore  a  side  of  a  regular 
hexagon. 


PROBLEM    I 

To  construct  an  angle  equal  to  a 
given  angle. 

Let  A  be  the  given  angle. 

With  A  as  a  center  and  a  radius  that  will 
cut  both  sides,  describe  the  arc  CD  ;  then, 
with  the  same  radius  and  B  as  a  center, 
describe  the  arc  EF,  making  it  equal  to  CD, 
and  draw  BF  ;  and  EBF  will  be  the  required 
angle. 

•PROBLEM    II.. 

Two  sides  and  the  included  angle  given,  to  construct 
a  triangle. 

Make  the  angle  A  equal  to  the  given 
angle,  and  on  one  side  lay  off  AB  equal 
to  one  of  the  given  sides,  and  on  the 
other  AC  equal  to  the  other  given  side. 
Draw  BC,  and  ABC  will  be  the  required  triangle. 

Cor. — Two  triangles  having  two  sides  and  the  included  angle 
respectively  equal,  are  equal  in  all  their  parts. 


30 


ELEMENTS     OF     GEOMETRY 


PROBLEM    III. 


One  side  and  the  two  adjacent  angles  given,  to  construct 
the  triangle. 

Make  AB  equal  to  the  given  side.  At 
A  construct  an  angle  equal  to  one  of  the 
given  angles,  and  at  B  an  angle  equal  to 
the  other  given  augle ;  the  intersection 
C  of  the  lines  forming  these  angles  will 

be  the  vertex  of  the  third  angle,  and  ABC  will  be  the  required 

triangle. 

Cor. — Two  triangles  having  each  a  side  and  the  two  adjacent 
angles  respectively  equal,  are  equal  in  all  their  parts. 


PROBLEM    IV. 

To  construct  a  triangle,  having  given  the  three  sides. 

Make  AB  equal  to  one  of  the  given 

sides.     Then  with  A  as  a  center,  and  a 

radius  equal  to  one  of  the  given  sides, 

describe  an  arc  ;  and  with   B  as  a  center 

and  the  other  given  side,  describe  an  arc  intersecting  the  other  arc 

at  C  and  draw  AC  and  BC,  then  ABC  will  be  the  required  triangle. 

Cor. — Two  triangles   having   their  three   sides  respectively 

equal,  are  equal  in  all  their  parts. 

Scho. — The  sum   of  any  two  sides  of  a  triangle  must  be 
greater  than  the  third  side. 


PROBLEM    V. 

Two  sides  and  an  angle  opposite  one  of  them  given, 
to  construct  a  triangle.  \ 

Make  the  angle  A  equal  to  the  given 
angle,  and  make  AC  equal  to  one  of  the 
given  sides;  then  with  C  as  a  center  and  a 
radius  equal  to  the  other  given  side,  draw 
the  arc  BB',  and  draw  CB  and  CB'.  In 
this  case  there  are  two  triangles. 


BOOK     II. 


31 


Scho. — The  second  side  must  be  equal  to  or  greater  than  the 
perpendicular  from  C  to  AB.  If  it  is  equal,  there  will  be  one 
right-angled  triangle ;  if  it  be  greater  than  the  perpendicular  and 
less  than  CA,  there  will  be  two  triangles;  but  if  it  be  less  than 
the  perpendicular,  there  will  be  no  triangle. 


PROBLEM    VI. 
Form  an  equilateral  triangle. 

Describe  a  circle,  and  apply  the  radius  six 
times  to  the  circumference,  and  draw  the 
chords;  the  result  is  a  hexagon.  Join  the 
alternate  vertices,  and  the  result  is  ABC,  an 
equilateral  triangle. 


PROBLEM    VII. 

To  construct  a  regular  polygon  of  eight  sides. 

Describe  a  circumference,  aud  divide  it 
into  eight  equal  parts.  Draw  chords  to  the 
equal  arcs ;  they  will  be  the  sides  of  the 
polygon. 

Draw  radii  from  the  extremities  of  the  sides 
to  the  center  of  the  circle ;  there  will  be  as 
many  isosceles  triangles  as  the  polygon  has 
sides.     The  angles  at  the  center  are  equal,  having  equal  arcs; 
and  each  angle  of  the  polygon  is  composed  of  two  equal  angles  of 
the  isosceles  triangles ;  hence,  all  the  angles  of  the  polygon  are 
equal ;  and  the  sides  being  also  equal,  the  polygon  is  regular. 

Cor. — A  regular  polygon  of  any  number  of  sides  may  be 
constructed  by  dividing  the  circumference  into  as  many  equal 
parts  as  there  are  sides. 

Rem. — The  circumference   will   be   divided  into   eight  equal 
parts  by  applying  the  chord  of  an  arc  of  45°. 


32 


ELEMENTS     OF     GEOMETRY, 


PROBLEM    VIII. 

To  draw  a  tangent  to  the  circumference  at  any  -point  on  it. 

Let  C  be  the  center  of  a  given  circle, 
and  A  any  point  in  its  circumference.  Draw 
the  radius  CA,  and  from  A  draw  AB  perpen- 
dicular to  the  radius  CA;  then  AB  will  be 
the  tangent  required.  (Book  1,  Th.  8, 
Cor.  2.) 

PROBLEM    IX. 

From  a  point  without  the  circle,  to  draw  a  tangent  to 
the  circle. 

Let  C  be  the  center  of  the  given  cir- 
cle, and  A  the  point  without  the  circle 
from  which  the  tangent  is  to  be  drawn. 
Join  the  point  A  and  the  center  C  and 
bisect  AC  in  0 ;  then,  with  0  as  a  cen- 
ter and  the  radius  OC  describe  a  cir- 
cumference ;  the  points  B  and  B',  the 
intersections  of  the  two  circumferences,  will  be  the  points  of 
tangency,  AB  and  AB'  the  tangents,  as  each  is  a  perpendicular  to 
a  radius  at  its  extremity;  the  angles  ABC  and  AB'C  being  each 
inscribed  in  a  semicircle. 


PROBLEM    X. 

On  a  straight  line,  to  construct  a  segment  that  shall 
contain  a  given  angle. 

Let  AB  be  the  given  line.  At  B  make 
the  angle  ABD  equal  to  the  given  angle. 
Draw  BO  perpendicular  to  BD,  and  at  C, 
the  middle  point  of  AB,  erect  a  perpendic- 
ular intersecting  BO  at  0  ;  then,  with  0  as 
a  center  and  radius  OB,  describe  a  circum- 
ference to  which  DB  is  a  tangent  and  AB  a 
chord,  and  the  angle  ABD  is  measured  by 
\  arc  AmB;  so  also  every  angle,  as  E,  E', 
inscribed  in  the  segment  AEB. 


BOOK     II. 

PROBLEM    XI. 

To  inscribe  a  circle  in  a  given  triangle. 

Bisect  any  two  angles  as  A  and 
B  by  the  straight  lines  AD  and  BD, 
and  as  every  point  in  each  bisecting 
line  is  equally  distant  from  the 
sides  of  the  angle,  hence  the  point 
of  intersection  D  will  be  equally 
distant  from  the  three  sides,  and  DE, 
DF  and  DG  will  be  radii  of  the  inscribed  circle. 


PROBLEM    XII. 

Draw  a  common  tangent  to  two  external  circles  of 
different  radii. 

Let  C  and  C  be  the  centers  of 
two  circles  which  are  external. 
With  C  as  a  center  and  a  radius 
equal  to  the  difference  of  the 
radii  of  the  circles,  describe  a 
small  circumference,  and  from  the 
point  C  draw  a  tangent  CA'  to 
this  small  circumference  ;    from 

the  center  C  draw  a  radius  through  the  point  of  tangency  A ,  and 
extend  it  to  A  in  the  circumference  of  the  large  circle  ;  draw  C'B 
parallel  to  CA  and  join  AB,  which  will  be  the  required  tangent. 

Cor. — A  second  tangent  B'D  may  always  be  drawn. 

PROBLEM    XIII. 

To  draw  a  tangent  to  two  external  circumferences  of 
different  radii,  the  tangent  passing  between  the  circles 
<$nd  touching  at  points  on  the  opposite  sides  of  the  cir- 
cumferences. 

Let  CA  and  C'B  be  the  radii  of  the 
given  circles.  With  C  as  a  center  and  a 
radius  equal  to  the  sum  of  the  two  given 
radii,  draw  a  circumference.  From  C 
draw  CD  and  CD'  tangents  to  the  large 
circle;  then  draw  the  radii  CD  and  CD', 

cutting  the  circumference  of  the  smaller  given  circle  in  A  and  A', 
which  will  be  the  points  of  tangency.  From  C  draw  C'B  parallel 
to  CA;  and  C'B'  parallel  to  CA',  and  join  AB  and  A'B',  and  they 
will  be  the  required  tangents. 


BOOK    III. 

PROPORTIONS 


DEFINITION". 

When  two  quantities,  each  having  the  form  of  a  fraction,  that 
is,  each  having  a  numerator  and  a  denominator,  are  equal  to  each 
other,  an  equation  may  be  formed  of  them ;  and  they  may  be 
arranged  proportionally.  In  order  to  show  whether  the  ratio  is 
increasing  or  decreasing,  the  denominators  should  be  made  the 
antecedents  and  the  numerators  consequents;  still,  they  are  in 
proportion  when  taken  in  an  inverse  order,  that  is,  the  numera- 
tors as  antecedents  and  the  denominators  as  consequents,  but  the 
ratios  will  be  inverted.     Thus, 


B 
A  = 

D 

"  c' 

Then  will 

A  :  B   : 

:   C 

:  D, 

and 

B  3-A   : 

:    D 

:  C. 

This  proportion 
solution;  thus, 

is  made 

10 
5   " 

very 

_  12 

6 

sim 

By  reduction, 

2 
1  " 

2 

end 

1:2: 

:    1  : 

2. 

The  same  proportion  as  5  :  10   : 

:   6  : 

12, 

or 

2:1: 

:   2  : 

1. 

and 

10  :  5   :: 

12  : 

C. 

simple  by   an  arithmetical 


The  equation  is  true  if  the  fractions  are  inverted ;  thus, 
56  a      l       ■*  O      1         o      -. 

15=5'    and    2  =  2;    •'•    *sl  ::  *-L 

Proportions  are  much  used  in  Geometry,  and  should  therefore 
be  carefully  studied. 

Instead  of  two  equal  ratios  there  may  be  many,  in  which  case 
they  are  termed  continued  proportions ;  as, 
B_D_F_H    _  K 
A  ~  C  ~  E : _  G  ~~  I 


etc. 


BOOK    III.  35 

Which  may  be  rendered, 

A  :  B   ::    C  :  D   ::    E  :  F   ::    G  :  H    ::    I  :  K,  etc. 

This  is  read:  as  A  is  to  B,  so  is  C  to  D,  so  is  E  to  F,  so  is  G  to  H, 
so  is  I  to  K.  The  antecedent  and  consequent  form  a  couplet,  and 
in  a  continued  proportion  any  two  couplets,  may  be  taken  to  form 
a  proportion  of  four  terms,  which  is  always  considered  a  propor- 
tion, and  the  first  and  last  terms  are  called  extremes,  and  the 
second  and  third  the  means. 


THEOEEM    I. 

If  four  quantities  are  proportional,  the  product  of  the 
means  equals  that  of  the  extremes. 

If  A  :  B   : :   C  :  D, 

a  B        D 

then  ^  =  -. 

Clearing  of  fractions  or  multiplying  both  members  by  A  and  C 
(Gen.  Ax.  6),  BC  =  AD. 

Cor.  1. — B  :  A  : :  D  :  C ;  that  is,  if  four  quantities  are  in 
proportion,  they  are  also  in  proportion  by  inversion. 

Cor.  2. — They  are  also  in  proportion  by  alternation  ;    thus, 

v-  =  p-     Multiplying  both  members  by  = ,  ■£=  =  p"r;  reducing, 

C        D 

j  —  =  ;  therefore,  A  :  C   : :    B  :  D,  and  again  by  inversion, 

C  :  A   ::    D  :  B. 


THEOREM    II. 

A  mean  proportional  between  two  quantities  is  equal 
to  the  square  root  of  their  product . 

Let  B  be  a  mean  proportional  between  A  and  C  ;  as, 

A  :  B  : :  B  :  C. 

The  product  of  the  means  is  equal  to  that  of  the  extremes; 
thus, 

B2  =  A  x  C. 
Extracting  the  root  of  both  members, 

B  =  VA~xC. 


36  ELEMENTS     OF     GEOMETRY. 


THEOREM    III. 

If  the  product  of  two  quantities  equals  the  product 
of  two  other  quantities,  either  of  the  two  forming  a 
product  may  be  made  the  means,  and  the  other  two  the 
extremes  of  a  proportion. 

Let  B  x  C  =  A  x  D; 

divide  by  A  x  C,  then 

B_x^  _  A_x^D  _  B  _  D 
AxC"AxC~A"C 
and  A  :  B    ::    C  :  D,  (1) 

or  C  :  D    ::    A  :  B.  (2) 

In  the  first  proportion,  A  and  D  are  the  extremes,  and  B  and 
C  the  means ;  in  the  second,  B  and  C  are  the  extremes,  and  A 
and  D  the  means. 


THEOREM    IV. 

If  four  quantities  are  proportional,  they  will  also  be 
proportional  by  composition  and  division. 

If    jf  =  £,     then    A-  +  l  =  -  +  l,    and    --1  =  ^-1. 

Reducing  to  improper  fractions, 


B  +  A       D  +  C  ,         B-A 

and 


A               C     '        A               C     ' 

and     A  :  B  +  A  ::  C  :  D  +  C;    also,  A  :  B— A  ::  C  :  D— C. 
By  alternation, 

A  :  C  ::  B  +  A  :  D  +  C;    also,  A  :  C  ::  B— A  :  D— C. 

C_D  +  C        .         C_  D-C 
A^BTA5    aUK>'    A"B^A" 

Gen.  Ax.  1,     ?±£  _  ^-C      .    B  +  A  :  D  +  C  ::  B-A  :  D-C. 
D  +  A         b — A 

By  alternation,         B  +  A  :  B  —  A    ::    D  +  C  :  D  —  C. 


BOOK     III.  37 


THEOREM    V. 

Like  powers  and  like  roots  of  proportional  quantities 
are  proportional, 

B        D  B2        D2 

Squaring  both  sides,        -r  =  ~  ;      then    ^  =  r2  > 

Bn       Dn  B»       D« 

and  x-  ss  7T-,      and      — r  =  — r-     (Gen.  Ax.  10.) 

A*       C'1'  A»       C» 

A2  :  B2    ::    C2  :  D2,      and       A»  :  Bn    ::    C  :  D», 

and  A»  :  B»    ::    O  :  D». 


THEOREM    VI 

Any  equimultiple  of  one  couplet  will  he  proportional 
to  the  other  couplet  or  to  any  equimultiple  of  it. 

This  depends  upon  the  principle  that  multiplying  both 
numerator  and  denominator  of  a  fraction  by  the  same  quantity 
does  not  change  its  value. 

A_c,       ana       m£-Q-nQ 


THEOREM   VII. 

If  the  corresponding  terms  of  two  proportions  be  mul- 
tiplied, their  products  will  be  proportional. 


hence  (Gen.  Ax.  6), 


A 

:  B    : 

:    C  : 

D 

E 

:  F    : 

:    G  : 

H 

B 
A  " 

D 

F 
E  ~ 

H 
=  G; 

BF 
AE  = 

DH 
"  CG* 

AE  : 

BF    : 

:    CG 

:  DH 

38  ELEMENTS     OF     GEOMETRY. 


THEOREM    VIII. 

In  a  series  of  proportions,  as  one  antecedent  is  to  its 
consequent,  so  is  the  sum  of  all  the  antecedents  to  the 
sum  of  all  the  consequents. 

A  :  B    ::    C  :  D    ::    E  :  F    ::    G  :  H,  etc. 

AD  =  BC 
AF  =  BE 
AH  =  BG 

AB  =  BA 

.      A(B  +  D  +  F-fH)  =  B(A  +  C+E  +  G) 

A  :  B    ::    A  +  C  +  E  +  G  :  B  +  D  +  F  +  H. 

Cor.  1. — If  any  two  proportions  have  an  equal  ratio,  then 
the  other  terms  are  proportional. 

Cor.  2. — The  same  is  true  if  the  antecedents  are  the  same  in 
two  proportions. 


BOOK     IT 


THEOKEM    I. 

The  area  of  a  rectangle  is  equal  to  the  product  of 
its  base  arid  altitude. 


There  may  be  three  cases : 

1st.  When  the  base  and  altitude  are 
composed  of  units  of  the  same  denomina- 
tion ;  then  it  is  evident  that  there  will  be 
as  many  square  units  for  every  unit  in  alti- 
tude as  there  are  units  in  the  base ;  and  for  every  additional  unit 
in  altitude  as  many  more  square  units  ;  hence  the  area  will  be 
the  product  of  the  base  and  altitude. 

2d.  If  there  be  a  fraction  in  one  or  both  the  dimensions,  the 
common  denominator  will  be  the  denomination  of  the  unit  of 
measure ;  hence,  the  product  of  the  base  and  altitude  will  give 
the  area,  in  units  of  the  same  denomination. 

3d.  If  the  dimensions  are  incommensurable,  the  unit  of 
measure  will  be  an  infinitesimal. . 


THEOREM    II. 

The  area  of  a  parallelogram  is  equal  to  the  product 
of  its  base  and  altitude. 

Let  ABCD   be  a  parallelogram,  AB  its  *» 

base,  BE  its  altitude,  its  area  =  AB  x  BE. 
Construct  the  rectangle  ABEF;  its  area  = 
AB  x  BE.     FE  =  AB  and  DC  =  AB,    .-.  FE       ALl  -JB 

=  DC,  and  taking  from  each  DE,  there 
remains  FD  =  EC;  hence  the  triangles  ADF  and  BCE  are  equal, 
having  all  their  sides  equal.  In  changing  the  parallelogram  into 
the  rectangle,  we  have  added  and  subtracted  the  same  area; 
hence  the  parallelogram  is  equal  to  the  rectangle.  .*.  the  area 
of  the  parallelogram  is  AB  x  BE,  product  of  base  and  altitude. 


4:0  ELEMENTS     OF     GEOMETBT, 


THEOREM    III. 

The  area  of  a  triangle  is  equal  to  one-half  the  pro- 
duct  of  the  base  and  altitude. 

Area  ABC  =  }(ABx  CE). 
Let  ABC  be  the  given  triangle,  AB  its 
base,  and  EC  its  altitude.  Construct  a 
parallelogram  on  AB  as  one  of  its  sides 
and  BC  as  another,  draw  AD  parallel  to 
BC  and  CD  parallel  to  AB;  then  will  ABCD  be  a  parallelogram. 
The  triangles  ABC  and  ACD  will  have  their  three  sides  respec- 
tively equal ;  hence  the  triangles  are  equal  and  each  is  one-half 
of  the  parallelogram  ABCD ;  and  as  the  area  of  the  parallelogram 
is  AB  x  CE,  that  of  the  triangle  is  \  (AB  x  CE) ;  therefore,  the 
area  of  a  triangle  is  equal  to  one-half  the  product  of  the  base  and 
altitude. 

Coe.  1. — Rectangles,  parallelograms  and  triangles  are  to  each 
other  as  the  products  of  their  bases  and  altitudes  respectively. 

Cor.  2. — If  the  bases  are  equal,  they  are  to  each  other  as  their 
altitudes. 

Cor.  3. — If  the  altitudes  are  equal,  they  are  to  each  other  as 
their  bases. 


THEOREM    IV. 

The  area  of  a  trapezoid  is  equal  to  the  product  of  its 
altitude  and  half  the  sum  of  its  parallel  bases. 

,c  Let  ABCD  be  a  trapezoid,  DE  its  alti- 

\  \  tude,  and  AB  and  DC  its   parallel   bases. 

\     \         Draw  the  diagonal  DB,  dividing  the  trape- 

\]B       zoid  into   two    triangles   whose    common 

altitude  is  DE  and  their  bases  AB  and  DC 


The  area  of  the  triangle  ABD  =  $  (AB  x  DE), 
"       "     "    "        "        BCD  =  j-(DC  x  DE), 
By  addition,  area  of      ABCD  =  DE  J-  (AB  +  DC). 

That  is,  the  area  of  a  trapezoid  is  equal  to  the  product  of  its 
altitude  and  \  the  sum  of  its  bases. 


BOOK 


41 


THEOREM    V. 

The  square  described  on  the  sum  of  two  lines,  is  equiv- 
alent to  the  sum  of  the  squares  of  the  lines,  increased  by 
twice  the  rectangle  of  the  lines. 

E  I  ^D 

ACDE'is  the  square  described  on  the  sum  of 
AB  and  BC;  and  corresponds  to  the  algebraic 
formula  {a  +  b)2  =  a2  +  2ab  -f-  b2,  in  which 
AB  =  a  and  BC  =  b. 


Cor. — If  the  lines  are  equal  there  will  be 
four  equal  squares.  Let  AB  =  1  and  BC  =  1  ; 
then  the  square  of  two  is  four  times  the  square 
of  one. 


ab 

ft  a 

H 

a* 

ab 

THEOREM    VI. 

The  square  described  on  the  difference  of  two  lines,  is 
equivalent  to  the  sum  of  the  squares  of  the  lines,  dimin- 
ished by  twice  the  rectangle  of  the  lines. 

AB  =  a,         KB  =  b,  a    b      e     d         _j c 

(a  —  b)2  =  a2  -  2ab  +  b2, 

ABCD  as  a\  EDFG  =  b\ 

BCIK  =  ab,     and    EIFH  =  ab, 


F      G  H 


THEOREM    VII. 

The  rectangle  contained  by  the  sum  and  difference  of 
two  lines,  is  equivalent  to  the  difference  of  their  squares. 

AB  =  a,        and        LB  =  BK  =  b, 
a  +  b  =  AK,     and    a  —  b  =  AE  =  AL, 
(a  +  b)  x  (a  —  b)  =  a2  —  b2, 
ABCD  =  a2,     and        FHGC  =  b2, 

A  L 

The  rectangle  EFGD  =  recti  BKIH  =  b  (a  —  b), 

ABHE  =  =  a  (a  —  b), 

By  addition,       AKIE  =  =  (a  +  b)  {a 


f; h 


B      K 


b). 


42 


ELEMENTS     OF     GEOMETEY. 


THEOREM    VIII. 

The  square  described  on  the  hypothenuse  of  a  right- 
angled  triangle,  is  equivalent  to  the  sujti  of  the  squares 
of  the  other  two  sides. 

Let  ABC  be  a  triangle,  right-angled  at  A, 

then  will  BC2  =  AB2  +  AC2, 

Construct  a  square  on  each  side  of  the 
triangle.  From  A  draw  a  perpendicular  to 
BC  and  extend  it  to  ED,  and  draw  AE,  AD, 
IC  and  BF.  The  triangles  ABE  and  IBC  have 
two  sides  respectively  equal,  viz.,  AB  =3  Bl 
and  BC  =3  BE,  being  respectively  sides  of  the 
same  square,  and  the  included  angles  equal ;  that  is,  ABE  =  I  BC, 
as  each  one  is  composed  of  the  angle  ABC  and  a  right  angle ; 
hence,  triangle  ABE  =  triangle  IBC;  but  triangle  ABE  is  one- 
half  the  rectangle  BELK,  having  the  same  base  and  altitude 
BE  and  BK  ;  and  the  triangle  IBC  is  one-half  the  square 
ABIH  =  IB  x  AB  =  AB2;  therefore,  AB2  =  rectangle  BELK. 

By  the  same  process,  we  prove  the  triangle  BCF  =  ACD,  and 
the  square  ACFG  ==  rect.  CDLK  ;  therefore,  BC2  =  AB2  +  AC2. 
And  by  transposing, 

Cor.  1.     BC2  -  AB2  =  AC2,  and  BC2  -  AC2  =  AB2. 

Cor.  2. — The  square  described  on  the  diagonal 
of  a  square  is  double  the  square  described  on  the 
side,  as  the  sides  are  equal ;  hence,  the  square  of 
diag.  :  sq.  of  side  : :  2:1,  and  diag.  :  side  : :  a/2  :  1. 
Since  AB2  =  rect.  BELK,  and  AC2  =  rect.  CDLK, 
the  resulting  proportion  AB2  :  AC2  ::  BK  :  KC;  that  is,  the 
squares  of  the  sides  are  proportional  to  their  adjacent  segments 


Cor.  3. 


of  the   hypothenuse. 


And  BC2  :  AB2 


BC  :  BK,  and  BC* 


AC  ::  BC  :  KC  ;  that  is,  the  square  of  the  hypothenuse  is  to 
the  square  of  either  side  as  the  hypothenuse  is  to  the  segment 
adjacent  to  the  side. 

Scho. — Observe,  that  if  the  right  angle  A  be  diminished,  the 
sides  about  it  remaining  the  same,  the  third  side  BC  will  be  dimin- 
ished; and  if  the  angle  A  be  increased,  BC  will  be  increased  ;  in 
the  first  case  the  square  of  BC  will  be  less,  and  in  the  second 
greater  than  the  sum  of  the  other  two  ;  hence,  the  right-angled 
triangle  is  the  only  one  in  which  the  square  of  one  side  is  equiv- 
alent to  the  sum  of  the  squares  of  the  two. 


BOOK     IV. 


43 


THEOREM    IX. 

In  any  triangle,  the  square  of  a  side  opposite  an  acute 
angle  is  equivalent  to  the  sum  of  the  squares  of  the  two 
other  sides,  minus  twice  the  rectangle  of  the  base,  and 
the  distance  from  the  acute  angle  to  the  foot  of  the  per- 
pendicular let  fall  from  the  vertical  angle  on  the  base, 
or  the  base  produced. 

In  the  triangle  ABC  the  side  AB  is  oppo- 
site the  acute  angle  C ;  hence, 

AB2  =  AC2  +  BC2  -  2BC  x  CD ; 

the  perpendicular  falling  on  the  base, 

BD  =  BC  -  DC. 
Squaring  both  members, 

_BD2  =  BC2  +  DC2  -  2B.C  x  DC, 
and  by  adding  AD2  to  each  member, 

BD2  +  AD2  =  BC2  +  DC2  +  AD2  -  2BC  x  DC; 
and  by  Theorem  8, 

AB2  =  BC2  +  AC2  -  2BC  x  DC. 

The  same  process  will  give  the  same  result, 
when  the  perpendicular  falls  upon  the  base 
produced. 


THEOEEM    X. 

In  an  obtuse- angled  triangle,  the  square  of  the  side 
opposite  the  obtuse  angle  is  equivalent  to  the  sum  of  the 
squares  of  the  two  other  sidles,  plus  twice  the  rectangle  of 
the  base  and  the  distance  of  the  obtuse  angle  from  the 
foot  of  the  perpendicular  let  fall  from  the  vertical  angle 
on  the  base  produced. 


Enunciation, 

AC2  =  AB2  +  BC2  +  2BC  x  BD, 
CD  =  BC+  BD; 
by  squaring, 

CD2  =  BC2  +  BD2  +  2BC  x  BD; 

A 

D         B 

adding  AD2  to  each  side, 

CD2  +  AD2  =  BC2  +  BD2  +  AD2  +  2BC  x  BD. 
Theorem  8.         AC2  =  BC2  +  AB2  +  2BC  x  BD. 


u 


ELEMENTS     OF     GEOMETRY, 


THEOEEM    XI. 

If  from  the  vertex  of  any  angle  of  a  triangle,  a  line 
be  drawn  to  the  middle  point  of  the  opposite  side,  then 
twice  the  square  of  the  bisecting  line,  plus  twice  the 
square  of  half  the  bisected  side,  will  be  equal  to  the  sum 
of  the  squares  of  the  two  other  sides. 

From   the  vertex   A  of  the  triangle 
'  ABC  draw  AD  to  the  middle  point  of  BC ; 
then  will 


D  2AD2  +  2BD2  =  AB2  +  AC2. 

In  the  triangle  ADC.,  the  side  AC  is  opposite  the  obtuse  angle  ADC. 

AC2  =  AD2  +  DC2  +  2DC  x  DE.  (1) 

And  in  the  triangle  AB  D  the  side  AB  is  opposite  the  acute  angle  AD  B. 

AB2  ■=  AD2  +  BD2  -  2BD  x  DE.  (2) 

By  adding  equations  (1)  and  (2),  and  observing  that  BD  =  DC, 
AB2  +  AC2  =  2AD2  +  2BD2. 

Cor. — The  sum  of  the  squares  of  all  the  sides  of  a  parallelo- 
gram, is  equivalent  to  the  sum  of  the  squares  of  the  diagonals. 
D _^0        Since    the   diagonals    mutually  bisect 


each  other, 


DC2  +  BC2 


By  addition,  AB 
(Th.  8,  Cor.  2.) 


2CE2  +  2DE2, 
AB2  +  AD2  =  2AE2  +  2DE2. 
+  DC2  +  AD2  +  BC2  =  4AE2  +  4DE2 


=    AC* 


BD2. 


THEOEEM    XII. 

If  a  line  be  drawn  parallel  to  one  of  the  sides  of  a 
triangle  cutting  the  other  sides,  it  will  divide  them  pro- 
portionally. 

Draw  DE  parallel  to  BC,  and  draw  BE  and 

DC;  then  the  triangles  DEB  and  DEC  have  the 

same  base  DE  and  the  same  altitude,  as  both 

their  vertices  are  in   the  line  BC,  parallel   to 

DE ;  hence,  they  are  equivalent. 

The  triangles  ADE  and  BDE  having  the  same  altitude,  as  they 

have  a  common  vertex  E,  are  to  each  other  as  their  bases ;  hence, 

ADE  :  BDE   ::   AD  :  BD. 


BOOK     IV. 


45 


The  triangles  ADE  and  DEC  have  a  common  vertex  D ;  hence, 

ADE  :  DEC   : :   AE  :  EC ; 

but  triangle  DEB  =  triangle  DEC,  and  the  two  proportions  have 
an  equal  ratio ; 

.-.    AD  :  BD   ::  AE  :  EC, 
and  by  composition, 


Cor.  1. 

AD  +  BD  :  BD  :: 

AE  +  EC  :  EC; 

that  is, 

AB  :  BD   :: 

AC  :  EC, 

and 

AD  +  BD  :  AD   :: 

AE  +  EC  :  AE, 

that  is, 

AB  :  AD  :: 

AC  :  AE. 

Cor.  2. 
parallel  to 
will  be  cut 

— If  any  number  of 
a  side  of  a  triangle, 
proportionally. 

lines   be  drawn 
the  other  sides 

i 

o 

Ax 

\f 

\d 

\ 

Cor.   3. — If  any  number  of  lines  be  cut  by 
the  parallels,  they  will  be  cut  proportionally. 


THEOREM    XIII, 


A  line  which  bisects  an  angle  of  a  triangle,  divides 
the  opposite  side  into  segments  proportional  to  the  adja- 
cent sides. 

Let  AD  bisect  the  angle  A ;  then 

BD  :  DC    ::    AB  :  AC.  (1) 

From  C  draw  a  line  parallel  to  DA,  inter- 
secting BA  produced  in  E ;  then 

BD  :  DC    ::'  AB  :  AE.  (2) 

The  angle  CAD  =  ACE,  alternate;  CAD  =  BAD,  bisected;  and 
BAD  =  BEC,  corresponding;  .*.  ACE  =  AEC,  and  the  triangle 
AEC  is  isosceles ;  hence,  side  AE  =z  AC,  and  (2), 

BD  :  DC    ::    AB  :  AC  =  AE. 


46 


ELEMENTS     OF     GEOMETRY. 


THEOREM    XIV. 

Triangles  which  are  mutually  equiangular  have  the 
sides  opposite  the  equal  angles  respectively  proportional, 
and  hence  the  triangles  are  called  similar. 

AB  :  DE  ::  AC  :  DF  ::  BC  :  EF. 
The  triangles  ABC  and  DEF  are 
mutually  equiangular ;  that  is,  A  =  D> 
E  =  B,  and  F  =  C.  Place  DEF  on 
ABC,  the  point  D  on  A,  the  side  DE 
on  AE',  and  DF  will  fall  on  AC,  since 
angle  D  =  angle  A.  Since  the  angles 
2  and  2  are  equal,  E'F'  is  parallel  to  BC;  and 

AB  :  AC    : :    AE'  :  AF', 
or,  AB  :  AC    ::    DE  :  DF; 

and  by  alternation,     AB  :  DE    ::    AC  :  DF. 
By  placing  F  on  C,  we  obtain  the  proportion, 

AC  :  DF    ::    BC  :  EF. 

Cor. — Two  triangles  having  two  angles  respectively  equal  are 
similar. 

Rem. — The  sides  opposite  the  equal  angles  are  called  homol- 
ogous. 


THEOREM    XV. 

The  figure  formed  by  joining  the  middle  points  of  any 
quadrilateral  by  straight  lines  is  a  parallelogram. 

Let  ABCD  be  any  quadrilateral. 
Join  the  middle  points  of  the  sides, 
and  draw  the  diagonals. 


BE 

:  BF  : 

:  EA  : 

FC 

AE 

:  AH  : 

:  EB 

:  HD 

DH  : 

DG  : 

:  HA 

GC 

CF  : 

CG  : 

:  FB 

GD. 

It  follows  that  GF  is  parallel  to  DB,  and  HE  is  also  parallel  to 
DB.     .-.  GF  and  HE  are  parallel ;  so  also  EF  and  HG  are  parallel. 


BOOK    IV.  47 


THEOREM    XVI. 

Two  triangles  which  have  their  sides  respectively  pro- 
portional are  similar. 


Since  the  sides  are  respectively  proportional,  then 

AB  :  DE    ::    AC  :   DF    ::    BC  :  EF  (1) 

Make   AE'  =  DE,    and  draw  E'F'  parallel  to  BC ;  then  will 

AB  :  AE'    ::    AC  :  AF'    ::     BC  :  E'F';  (2) 

but  AE'  =  DE. 

The  proportions  (1)  and  (2)  have  an  equal  ratio,  hence  the 
other  ratios  must  be  the  same ;  hence 

AF'  ==  DF 

and  E'F'  =  EF; 

the  triangle  AE'F'  =  DEF; 

but  the  triangles  AE'F'   and   ABC   are  equiangular,  as  E'F'  is 
parallel  to  BC ;  and  as  they  are  equiangular  they  are  similar. 

Cor.  1. — If  two  triangles  have  each  an  equal  angle  included 
by  proportional  sides,  they  are  similar. 

Cor.  2. — Two  triangles  which  have  their  sides  respectively 
parallel  or  perpendicular  to  each  other  are  similar. 


48 


ELEMENTS     OF     GEOMETRY. 


THEOREM    XVII. 

Regular  polygons  of  the  same  number  of  sides  are 
similar  figures. 

Construct  a  regular  polygon,  as  in 
Problem  VII,  Book  II,  and  let  a  smaller 
one  be  placed  upon  it,  the  angles  being 
the  same  in  both  polygons  ;  they  will  also 
be  the  same  in  the  isosceles  triangles; 
consequently  the  sides  AB  and  ab,  also  BC 
and  be,  etc.,  will  be  parallel ;  hence  the 
proportions, 

AB  :  ab    ::    R  :  r, 

BC  :  be    ::    R  :  r,    etc. 

.-.  the  triangles  are  similar;  and  as  each  polygon  is  composed 
of  an  equal  number  of  similar  triangles,  the  polygons  are 
similar. 

Cor.  1. — It  is  evident  that  a  circumference  may  be  inscribed 
in  the  polygon,  as  the  perpendicular  OP,  which  is  called  the 
apothegm  of  the  polygon  is  the  distance  from  the  center  to  each 
side. 

Cor.  2. — As  the  equal  sides  of  the  isosceles  triangles  become 
radii  of  the  circumscribed  circle,  a  circle  may  be  passed  through 
all  the  vertices. 

Cor.  3. — Circles  are  similar  figures,. 


Def. — Two  polygons  which   are  mutually  equiangular  and 
have  their  corresponding  sides  proportional,  are  similar. 


BOOK     IV. 


49 


THEOREM    XVIII. 

In  a  right-angled  triangle,  if  aline  be  drawn  from 
t-he  right  angle  perpendicular  to  the  hypothenuse,  it 
will  divide  the  given  triangle  into  two  triangles,  simi- 
lar to  the  given  triangle  and  similar  to  each  other. 

Let  ABC  be  right-angled  at  C,  and 
C  D  perpendicular  to  the  hypothenuse  AB. 
The  triangles  ABC  and  ADC  have  the 
angle  A  common,  and  each  has  a  right 
angle;  they  are  therefore  similar.  And 
for  the  same  reason  ABC  and  BCD  are  similar;  consequently, 
ADC  and  BCD  are  similar. 


Cor.  1.     Since  ABC  and  ADC  are  similar, 
AB  :  AC    ::    AC  :  AD, 
AB  x  AD  =  AC2.  .  (1) 

Since  ABC  and  BCD  are  similar, 

AB  :  BC   ::    BC  :  BD, 
AB  x  BD  =  BC2.  (2) 

Since  ADC  and  BCD  are  similar, 

AD  :  DC   ::    DC  :  BD;  (3) 

whence,  DC2  =  AD  x  BD. 

Adding  (1)  and  (2),       AB2  =  AC2  +  BC2. 
From  proportions  (1)  and  (2),  the  result  is  that  the  square  of 
the  hypothenuse  is  equivalent  to  the  sum  of  the  squares  of  the 
other  sides  ;  and  from  (3),  that  the  perpendicular  is  a  mean  pro- 
portional between  the  segments  of  the  hypothenuse. 

Cor.  2.  If  from  any  point  in  the  cir- 
cumference of  a  circle  a  perpendicular  be 
drawn  to  the  diameter,  it  will  be  a  mean 
proportional  between  the  segments  of  the 
diameter. 

Let  C  be  the  point  in  the  circumfer- 
ence from  which  is  drawn  the  perpendic- 
ular CD  to  the  diameter  AB;    by_drawing  AC  and  CB,  ABC 
becomes  right-angled  at  C;  hence,  CD2  =  AD  x  BD.     (3) 
3 


50 


ELEMENTS     OF     GEOMETRY, 


THEOREM    XIX. 

If  two  chords  intersect  each  other  in  a  circle,  their  seg- 
ments are  reciprocally  proportional. 

The  triangles  ACE  and  BCD  are  simi- 
lar; the  angle  E  =  angle  B,  and  A  =  D, 
respectively  measured  by  \  the  same  arc ; 
hence,    AC  :  DC    ::    CE  :  BC. 

Cor.  AC  x  BC  =  DC  x  CE,  the 
product  of  the  segments  of  the  one  chord 
equal  to  the  product  of  the  segments  of 
the  other  chord. 


THEOREM    XX. 

If  from  a  point  without  a  circle,  two  secants  be  drawn 
terminating  in  the  concave  arc,  the  whole  secants  will  be 
reciprocally  proportional  to  their  external  segments. 

In  the  similar  triangles  ACE 
and  BCD, 

AC  :  BC    ::    CE  :  DC. 
Cor.    AC  x  DC  =  BC  x  CE. 


THEOREM    XXI. 

If  from  a  point  ivithout  the  circle  a  tangent  and  a 
secant  be  drawn,  the  tangent  will  be  a  mean  proportional 
between  the  whole  secant  and  its  external  segment. 

The  similar  triangles  ABC 
and  ACD  give  the  following  pro- 
portion : 


CB  :  AC    :: 
hence,  CB  xCD 


AC  :  CD; 

:AC2. 


BOOK     IV. 


51 


THEOREM    XXII. 

i"/  a  line  be  drawn  bisecting  an  angle  of  a  triangle  and 
intersecting  the  opposite  side,  the  rectangle  of  the  sides 
about  the  bisected  angle  equals  the  rectangle  of  the  seg~ 
ments  of  the  third  side  plus  the  square  of  the  bisecting 
line. 

Circumscribe  a  circle  about  the  given 
triangle  ABC,  and  bisect  the  angle  C  and 
extend  the  bisecting  line  to  the  circumfer- 
ence of  the  circle  and  draw  BE.  The 
triangles  ADC  and  BCE  are  similar;  hence, 

AC  :  CE    ::    CD  :  BC; 

AC  x  BC  =  CE  x  CD; 
but  CE  =  CD  +  DE, 

and  (CD  +  DE)  x  CD  =  DE  x  CD  +  CD2, 

and  DE  x  DC  =  AD  x  DB. 

AC  x  BC  =  AD  x  BD  +  CD2. 


THEOREM    XXIII. 

Two  triangles,  having  an  angle  in  each  equal,  are  to 
each  other  as  the  rectangles  of  the  sides  containing  the 
equal  angles. 


Let  the  triangles  ABC  and  ADE  have  the  angle 
A  common;  then  will  ABC  :  ADE  ::  AB  x  AC  : 
AD  x  AE.  Draw  BE;  then  the  triangles  ABE 
and  ADE  have  the  same  altitude,  and  hence  ABE  : 
ADE  ::  AB  :  AD;  the  triangles  ABC  and  ABE 
have  the  same  altitude,  ABC  :  ABE  ::  AC  :  AE. 
Multiplying  the  proportions  and  observing  that 
ABE  is  common  to  antecedent  and  consequent, 

ABC  :  ADE   ::   AB  x  AC  :  AD  x  AE. 


52 


ELEMENTS     OF     GEOMETKY. 


THEOREM    XXIV. 

Similar  triangles  are  to  each  other  as  the  squares  of 
their  homologous  sides. 

Let  ABC  and  DEF  be  similar  triangles ; 
angle  A  =  angle  D, 

ABC  :  DEF  ::  AB  x  AC  :  DE  x  DF.  (1) 
AB  :   DE   ::   AC  :  DF.  (2) 

Multiplying  this  proportion  by  the  identical  pro- 
portion, 

DF  ::   AC  :  DF, 


AC 
AB  x  AC 
Since  the  1st  and  4th  have  equal  ratios 


DE  x  DF  : :  AC2  :  DF2- 


(3) 

(4) 


ABC  :  DEF   ::   AC2  :  DP 


and  as  the  homologous  sides  are  proportional,  so  also  the  triangles 
are  to  each  other  as 

AB2  :  DE2    and     BC2  :  EF2- 

Ooe,  1. — The  areas  of  regular  polygons  are  to  each  other  as 
the  squares  of  the  radii  of  the  inscribed  or  circumscribed  circle. 

Cor.  2. — The  areas  of  circles  are  to  each  other  as  the  squares 
of  their  radii,  or  the  squares  of  the  diameters. 


GENERAL    COROLLARIES. 

1.  The  perimeters  of  similar  polygons  are  to  each  other  as 
their  homologous  sides,  or  as  their  corresponding  diagonals. 

2.  The  perimeters  of  regular  polygons  of  the  same  number  of 
sides  are  to  each  other  as  the  radii  of  the  inscribed  or  circum- 
scribed circles. 

3.  The  circumferences  of  circles  are  to  each  other  as  their 
radii  or  diameters. 


BOOK     IV, 


53 


g   f 


PROBLEM    I. 

To  divide  a  given  line  into  five  equal  parts* 

Let  AB  be  the  given  line.  From  A 
draw  an  indefinite  line  AH,  making  any 
angle  with  AB,  and  on  it  lay  off  the 
same  distance  five  times.  Join  the  last 
point  C  with  B,  and  from  each  point 
draw  lines  parallel  to  CB;  then  AB  will  be  divided  into  five  equal 
parts.     (Th.  12,  Cor.  2.) 

PROBLEM    II. 

To  divide  a  given  line  into  parts  proportional  to  several 
given  lines. 

Let  AB  be  the  given  line,  abed 
Draw  AH  an  indefinite  line,  and 
on  it  lay  off  the  several  given 
lines  a,  b,  c,  d,  and  join  the  last 
point  with  B,  and  from  each  point 
draw  lines  parallel  to  this  line;  then  will  the  parts  Aa',  ab',  b'c, 
and  c'B  be  proportional  to  the  given  lines  a,  b,  c,  and  d. 


ab  e 


PROBLEM    III. 

To  find  a  fourth  proportional  to  three  given  lines. 

From  any  point,  as  A,  draw  two 
indefinite  lines  AH  and  AY ;  on  AY  lay 
off  a,  and  on  AH  lay  off  b  and  join  ab; 
on  AY  lay  off  c  and  draw  cd  parallel  to 
ab ;  bd  will  be  a  fourth  proportional. 


PROBLEM    IV. 
To  construct  a  mean  proportional  to  two  given  lines. 

On  an  indefinite  line  lay  off  AB  and  BC, 
equal  respectively  to  the  given  lines.  On 
AC  describe  a  semi-circumference,  and  at  B 
erect  the  perpendicular  BD,  which  will  be 
a  mean  proportional  between  AB  and  BC. 
(Th.  18,  Cor.  2.) 


54 


ELEMENTS     OF     GEOMETRY, 


PROBLEM    V. 

To  construct  a  triangle  equivalent  to  a  given  -polygon. 

Let  ABCDE  be  the  given  polygon.  From 
A  draw  the  diagonals  AD  and  AC ;  then  from 
E  and  B  draw  EF  and  BG,  respectively  par- 
allel to  the  diagonals  AD  and  AC,  intersect- 
ing the  base  produced ;  then  AFG  will  be 
the  required  triangle. 


c 


PEOBLEM    VI. 

To  inscribe  a  square  in  a  circle  and  circumscHbe  a 
square  about  a  circle. 

Draw  two  diameters  at  right  angles  and 
join  their  extremities,  and  we  have  an  inscribed 
square,  as  each  side  is  a  chord  of  ninety  de- 
grees, and  each  angle  is  measured  by  one-half  a 
semi-circumference.  At  each  extremity  of  the 
perpendicular  diameters  draw  tangents  to  the 
circumference  and  we  have  the  circumscribed 
square. 

Cor.  1. — The  circumscribed  square  has  double  the  area  of  the 
inscribed,  as  it  has  eight  equal  triangles,  whilst  the  inscribed  has 
only  four  of  the  equal  triangles  ;  hence,  area  of  cir.  sq.  :  area  of 
ins.  sq.  ::  2:1,  and  side  of  cir.  sq.  :  side  of  ins.  sq.  ::  \/2  :  1; 
same  result  as  in  Th.  8,  Cor  2. 

Scho. — The  side  of  the  circumscribed  square  is  the  same  as  the 
diagonal  of  the  inscribed. 

Cor.  2,— In   the  triangle    CBC,  right- 
angled  at  B,  we  have 

CC72  -  CB2  =s  CB2. 

CC  =  2,         and         CB  as  1. 

4  -  1  =  CB2, 

and  CB2  =  3, 

CB   =  V3. 

That  is,  TJie  side  of  an  equilateral  triangle  :  Radius  : :  \/3  :  1. 


BOOK     IV. 


55 


PROBLEM  VII. 

To  divide  a  given  line  into  extreme  and  mean  ratio ; 
that  is,  into  two  such  parts  that  the  greater  part  shall  be  a 
mean  proportional  between  the  whole  line  and  the  less  part. 

Let  AB  be  the  given  line.     At  B  erect  a 
perpendicular  BC  equal  to  -JAB ;  and  with  C 
as  a  center  and  radius  CB,  describe  a  circum- 
ference.    From  A  draw  AF  through  the  center 
and  terminating  in  the  concave  arc,  and  with 
A  as  a  center  and  AD  as  radius,  draw  the  arc 
DE,  making  AE  equal  to  AD ;  then  DF  =  AB,  and  (Theorem  21) 
AF  :  AB   ::   AB  :  AD,  by  inversion  AB  :  AF   ::   AD  :  AB,  and  by 
division  AB  :  AF  -  AB  ::   AD  :  AB  -  AD;   that  is,  AB  :  AD   :: 
AD  :  EB,  orAB  :  AE   ::   AE  :  EB. 


PROBLEM    VIII. 

To  construct  a  square  equivalent  to  a  given  triangle. 

A  mean  proportional  between  the    BAe*-  l  altitudb. 
base  and  half  the  altitude  of  the  tri- 
angle will  be  a  side  of  the  square. 

Let  B  =  Base  and  A  =  \  Altitude. 
DC  is  a  mean  proportional  between 
base  and  one-half  altitude. 


*  PROBLEM    IX. 

To  construct  a  square  equivalent  to  two  given  squares. 

Construct  a  right  angle.  On  one  of  a  b 
the  sides  of  the  angle  lay  off  a  distance 
equal  to  a  side  of  one  of  the  squares ;  and 
on  the  other  side  of  the  angle  a  distance 
equal  to  a  side  of  the  other  square,  and 
draw  the  hypothenuse ;  it  will  be  a  side 
of  the  required  square. 

Rem. — By  this  principle  the  side  of  a  square  equivalent  to 
any  number  of  squares  may  be  found. 

Cor. — By  making  the  longer  side  the  hypothenuse,  the  third 
side  will  be  the  side  of  a  square  equal  to  the  difference  of  two 
squares. 


56 


ELEMENTS     OF     GEOMETRY. 


Rem.  1. — If  similar  polygons  be  constructed  on  the  three  sides 
of  a  right-angled  triangle,  the  given  sides  being  homologous,  the 
polygon  constructed  on  the  hypothenuse  will  be  equivalent  to  the 
sum  of  the  two  others. 

Rem.  2. — To  construct  a  square  equivalent  to  a  given  poly- 
gon, reduce  the  polygon  to  an  equivalent  triangle  and  find  a 
mean  proportional  between  the  base  and  half  the  altitude  of  the 
triangle. 

PROBLEM    X. 

To  construct  a  polygon,  similar  to  a  given  polygon,  on  a 
given  side  homologous  to  one  of  the  sides  of  the  given 
polygon. 

Let  ABCDEF  be  the  given  polygon 
and  AB'  a  side  of  the  required  polygon 
homologous  to  AB.  Lay  off  AB'  on  AB, 
and  from  A  draw  all  the  diagonals. 
Draw  B'C  parallel  to  BC  to  the  first 
diagonal ;  then  from  one  diagonal  to 
another  draw  sides  parallel  to  the  opposite  side  of  the  given  poly- 
gon. AB'C'D'E'F'  will  be  the  required  polygon.  (Th.  17,  Cor.  4.) 
Cor. — To  construct  a  regular  polygon,  having  one  of  the 
sides  given:  First  construct  a  regular  polygon  of  the  proper 
number  of  sides;  then  find  a  fourth  proportional  to  the  side  of 
the  constructed  polygon,  the  side  of  the  required  polygon,  and 
the  radius  of  the  circumscribed  circle  of  the  constructed  polygon ; 
the  fourth  proportional  will  be  the  radius  of  the  circumscribed 
circle  of  the  required  polygon. 


PROBLEM    XI. 

To  extract  the  square  root  of  a  quantity,  or,  what  is  the 
same  thing,  to  find  the  side  of  a  square  equivalent  to  a 
given  surface. 


The  surface  of  a  square  is  found  by  squaring 
a  side;  thus,  3x3  =  9,  that  is,  3  in  length  and 
3  in  breadth.  9  is  the  surface  of  which  we  wish 
to  find  a  side  of  a  square  equivalent ;  and,  as 
3  x  3  =  9,  it  is  evident  that  3  is  the  square  root 
of  9  ;  so  also  4  of  16,  5  of  25,  6  of  36,  etc. ;  but 
when  the  number  is  large  it  is  not  so  easily  found. 


BOOK     IV. 


5? 


Let  us  take  an  algebraic  binomial,  as  (a  +  b)2 
=  a2  +  2ab  -\-  b2,  and  exhibit  it  geometrically. 

The  divisors  must  be  such  as  to  render  the 
quotient  a  root  (a  +  b). 

a)  a2  +  2ab  +  Vi(a  +  b 


ab 

b2 

a2 

ab 

2a  +  b  )   +  2ab  +  b2 

+  2ab  +  W> 


Next  take  a  trinomial ;  as 
a  )  a2  +  2ab  +  b2  +  2ac+2bc+c2  (  a  +  5  +  c 


2a  +  b  )  2«6  +  £2 
2^  +  52 


2a  +  2£-fc)  -f-2ffc  +  2fc+c* 

+  2«c  +  2^4-6'2 


M 

?H- 

,•2 

a& 

/;2 

^ 

«2 

aZ> 

■3 

Cor.  1. — The  first  term  of  the  roots  is  obtained  the  same  as 
that  of  finding  the  root  of  a  monomial;  the  divisor  and  root  are 
the  same,  as  the  first  surface  a2  is  a  square ;  after  that  we  have 
rectangles,  the  breadth  of  which  is  the  root,  and  our  divisor  must 
be  the  whole  length  of  the  rectangles. 

Cor.  2. — Each  successive  divisor  is  double  the  root  already 
found,  plus  the  next  term  of  the  root. 

Rem. — In  Arithmetic  we  pursue  the  same  course,  except 
that  the  squares  are  not  so  entire  and  separate  as  in  Algebra  and 
Geometry;  hence,  in  general,  we  take  the  nearest  root,  the 
largest  figure  of  which  the  square  is  less  than  the  given  number, 
and  we  point  off  the  figures  in  periods  of  two  each,  beginning  at 
the  unit's  place.  The  reason  of  pointing  off  in  periods  is  shown 
by  the  increase  of  the  numbers  in  squares ;  thus, 


11 

11 

121 


9 
_9 
81 


99 
99 


9801 


The  increase  of  one  figure  in  the  side  makes  two  in  the  sur- 
face ;  it  will  always  be  this,  and  never  more  or  less,  as  is  shown 
by  taking  the  smallest  and  the  largest  digits. 


58 


ELEMENTS     OF     GEOMETRY. 


PROBLEM    XII. 

To  find  the  circumference  of  a  circle  whose  radius  is 
unity. 

With  1  as  a  radius  describe  a  circumfer- 
ence, and  inscribe  in  it  a  regular  hexagon, 
each  side  of  which  will  be  unity.  Take  any 
side,  as  AB,  and  bisect  it  in  D  and  its  arc  in 
C,  and  draw  the  chord  CB,  which  will  be 
the  side  of  a  regular  polygon  of  double  the 
number  of  sides.  The  triangle  ODB  is  right- 
angled  at  D ;  hence, 

OD  =  Vo B2  -  61?  =  Vl^i  ==  l-y/3, 

and        CB  =  Vj  +  (l 


and        CD  =  1—  |V3, 

Let  C  represent  a  side  of  the  first  polygon,  and 


iV3)2. 
c  a  side  of  the 


polygon  of  double  the  number  of  sides  ;  in  each  successive  com- 
putation, after  c  is  found,  make  it  C  in  the  next,  and  continue 
this  process  until  the  difference  between  C  and  c  has  no  appre- 
ciable value ;  then  this  value  of  C  multiplied  by  the  number  of 
sides  will  give  6.2832,  which  is  the  approximate  length  of  the 
circumference  when  the  radius  is  1  and  the  diameter  2. 

When  the  diameter  is  1,  the  circumference  is  3.1416,  which 
number  is  represented  by  tt  ;  hence,  nd  or  2nr  represents  circum- 
ference. 

PROBLEM    XIII. 

The  area  of  a  regular  -polygon  is  equal  to  its  perimeter 
multiplied  by  one-half  the  radius  of  the  inscribed  circle. 

Let  ABCDEF  be  a  regular  inscribed 
hexagonal  polygon,  and  OK  the  radius  of 
the  inscribed  circle.  The  polygon  is  com- 
posed of  six  triangles,  each  having  a  side 
of  the  polygon  for  its  base  and  OK  for  its 
altitude ;  hence  the  area  of  the  polygon  is 
its  perimeter  multiplied  by  £OK;  that  is, 
\  the  radius  of  the  inscribed  circle. 

Coe. — When  the  number  of  sides  of  the  polygon  is  indefinitely  increased, 
it  becomes  a  circle,  and  the  radius  of  the  inscribed  circle,  which  has  been 
increasing  as  the  number  of  sides  increased,  is  now  the  radius  of  the  circle, 
and  the  perimeter  of  the  polygon  is  the  circumference  of  the  circle  ;  hence, 
the  area  of  a  circle  is  equal  to  its  circumference  multiplied  by  one-half  its 
radius. 


BOOK    V. 


THEOEEM    I. 

When  the  distance  between  the  centers  of  two  circles  is 
greater  than  the  sum  of  their  radii,  they  are  external,  and 
the  straight  line  joining  their  centers  will  be  the  shortest 
distance  between  the  center  of  either  circle  and  the  circum- 
ference of  the  other ;  and  if  this  line  be  extended  to  bite 
concave  arcs  of  both  circles,  it  will  be  greater  than  between 
any  two  other  points  in  the  circumference. 

Let  C  and  C  be  the  centers  of  two  circles  external  to  each 
other;  C  B  is  the  shortest  distance  from  C  to  any  point  in  the 


circumference  C  ;  for,  let  the  tangents  DE  and  D'E'  be  drawn  at 
B  and  A',  they  will  be  perpendicular  to  C'C  ;  and  as  a  perpendic- 
ular is  the  shortest  distance  from  a  point  to  a  line,  C'B  is  the 
shortest  distance  from  C  to  the  tangent  DE,  and  any  other  line 
from  C  to  the  circumference  is  oblique  to  the  line  DE,  and  must 
go  beyond  it  before  it  can  reach  the  circumference. 

2.  C'A  is  longer  than  any  other  line  drawn  from  C  to 
the  circumference  C,  as  CF.  Draw  the  chord  BF;  BF  is  less 
than  AB  a  diameter;  hence, 

C'B  -r  BF  <  C'A, 

and  CT<C'B+BF; 

much  more  then  is  C'A  >  CF. 


60 


ELEMENTS     OF     GEOMETRY. 


THEOEEM    II. 

When  the  distance  between  the  centers  is  equal  to  the 
sum  of  the  radii,  they  are  tangents  externally ,  and  the 
straight  line  joining  their  centers  passes  through  the 
point  of  tangency. 

They  must  touch  on  the  line  joining 
their  centers,  as  CD  +  DC  =  CC;  let  D 
be  this  point,  and  through  D  draw  AB  per- 
pendicular to  CC;  it  will  be  a  common 
tangent  to  both  circles  as  it  is  perpendic- 
ular to  each  radius  at  its  extremity. 

Or,  CC  is  the  shortest  distance  between  the  centers,  and  CD  is 
the  shortest  distance  from  C  to  AB ;  CD  is  also  the  shortest 
distance  from  C  to  AB  ;  therefore,  the  line  between  the  centers 
of  the  two  circles  passes  through  their  point  of  tangency. 


THEOEEM    III. 

When  the  distance  between  the  centers  is  less  than  the 
sum  and  greater  than  the  difference  of  the  radii,  they 
intersect  each  other,  and  the  line  joining  their  centers  is 
perpendicular  to  their  common  chord  at  its  middle  point. 

CC  is  the  shortest  distance  between  C 
and  C;  hence,  CD  and  CD  must  both  be 
perpendicular  to  A  B  at  its  middle  point,  and 
CC  must  be  a  straight  line. 


THEOEEM    IV. 

When  the  distance  between  the  centers  is  equal  to  the 
difference  of  the  radii,  the  smaller  circle  is  tangent 
internally  to  the  larger  one,  and  the  line  joinijig  their 
centers  extended  passes  through  the  point  of  ta:igency. 


C'A  is  the  shortest  distance  from  the 
center  C  to  the  circumference  C ;  therefore, 
A  is  the  point  of  tangency. 


BOOK     V 


61 


THEOEEM    V. 

When  the  distance  between  the  centers  is  less  than  the 
difference  of  the  radii,  the  smaller  is  wholly  within  the 
larger,  and  the  nearest  and  the  fartJiest  points  in  the 
circumference  of  the  one  circle,  from  the  center  of  the  other 
circle,  is  in  the  extensions  of  the  line  joining  their 
centers. 


C'B  is  the  nearest  distance  and  C'A  is 
the  farthest  in  the  circumference  C  from  the 
center  C     (Th.  1.) 

Cor. — When  they  are  concentric,  their 
circumferences  are  parallel. 


Gen.  Cor. — The  line  joining  the  centers  passes  through  the 
points  of  taiigency,  the  middle  point  of  the  common  chord,  the 
nearest  and  the  farthest  point  of  the  circumference. 


THEOREM    VI 


If  on  one  side  of  a  given  polygon  another  polygon  be 
constructed  within  the  given  polygon,  the  perimeter  of  the 
interior  polygon  will  be  less  than  that  of  the  given 
polygon. 

Produce  each  side  of  the  interior  poly- 
gon until  it  meets  a  side  of  the  exterior. 
Then  ke  is  less  than  A  Be, 

If  <  beCf,  eg  <  cfDg,  and  dE  <  dgE. 

.-.    AeCDE<ABCDE;     A5/DE<A*CDE; 

kbcgE  <  AJ/DE ;    and    klcdE  <  A%E ; 

much  more  is  klcdE  less  than  ABCDE. 

Cor. — If  from  any  point  within  a  triangle  lines  be  drawn  to 
the  extremities  of  either  side,  the  sum  of  these  lines  will  be  less 
than  the  sum  of  the  two  other  sides. 


62 


ELEMENTS     OF     GEOMETRY. 


THEOREM    VII. 

If  two  circles  whose  radii  are  unequal  intersect  each 
other,  the  middle  point  of  their  common  chord  will  be 
nearer  the  arc  of  the  large  circle  than  that  of  the  smaller 
one. 

Let  C  and  C  be  the  centers  of  the  two  circles ;  then  will  DF 
be  less  than  DE.  For  place  the  two  circles  so  that  the  smaller 
one  shall  be  tangent  internally  to  the  larger  one. 


Draw  AB  a  chord  to  the  larger  circle,  A'B'  will  be  a  chord  of 
the  smaller  one  at  the  same  distance  from  the  arcs  of  the  circles  ; 
the  chord  AB  is  longer  than  the  chord  A'B',  and  in  order  that 
A'B'  become  equal  to  AB  it  must  be  put  nearer  the  center  of  the 
circle  and  farther  from  the  arc  ;  therefore,  DF  is  less  than  DE. 


THEOREM    VIII 

If  the  circumferences  of  two  unequal  circles  intersect 
each  other,  the  arc  of  the  larger  circle  is  less  than  that 
of  the  smaller  one. 

Let  the  circle  C  be  larger  than  C,  and 
let  their  circumferences  intersect  at  A  and 
B ;  then  will  the  arc  AFB  of  the  larger 
circle  be  less  than  the  arc  AEB  of  the 
smaller  one. 

Since  DF  is  less  than  DE  (Th.  7),  the. arc  AFB  may  be 
revolved  on  AB  as  an  axis,  the  point  F  will  fall  upon  ED,  between 
D  and  E,  and  the  arc  AFB  will  be  wholly  within  the  arc  AEB, 
and  is  therefore  less.     (Th.  6.) 

Rem. — An  arc  may  be  regarded  as  a  portion  of  the  perimeter 
of  a  polygon. 


BOOK    VI. 

PLANES     IN     DIFFERENT     POSITIONS. 


DEFINITIONS. 

1.  A  straight  line  is  perpendicular  to  a  plane,  when 
it  is  perpendicular  to  every  line  passing  through  its  foot  in  that 
plane. 

2.  Two  planes  are  parallel  when  they  are  everywhere 
equally  distant,  and  consequently  will  never  meet. 

3.  Three  points  not  in  the  same  straight  line  determine 
the  position  of  a  plane. 

4.  The  intersection  of  two  planes  is  a  straight  line, 
for  two  planes  cannot  have  three  points  common  which  are  not 
in  the  same  straight  line ;  for,  if  they  have,  they  form  but  one 
plane. 

5.  A  Diedral  Angle  is  the  divergence  of  two  planes  from 
their  common  intersection,  and  is  measured  by  the  plane  angle 
formed  by  two  lines,  one  in  each  plane,  perpendicular  to  the 
common  intersection,  at  any  point  in  this  line. 

Rem. — Theorems  regarding  several  planes  in  different  posi- 
tions have  no  distinct  principles  ;  but  special  attention  is  required 
to  the  position  of  the  planes,  as  a  correct  figure  of  two  or  more 
planes  cannot  be  drawn  upon  one  plane,  as  the  blackboard. 

The  few  theorems  given  on  this  subject  are  thought  sufficient 
to  illustrate  this  peculiarity. 


64 


ELEMENTS     OF     GEOMETRY. 


THEOREM    I. 

If  from  a  point  without  a  plane,  a  perpendicular  be 
drawn  to  the  plane  and  oblique  lines  to  different  points  of 
the  plane, 

1st.  Any  oblique  line  will  be  longer  than  the  perpen- 
dicular. 

2d.  Oblique  lines  drawn  to  points  equally  distant  from 
the  foot  of  the  perpendicular  ivill  be  equal. 

3d.  Oblique  lines  unequally  distant  from  the  foot  of  the 
perpendicular,  the  one  farther  distant  will  be  the  longer. 

Let  P  be  a  point  without  the  plane 
MN,  PC  a  perpendicular  to  the  plane, 
and  A  and  B  two  points  in  the  plane  M  N 
equally  distant  from  C  the  foot  of  the 
perpendicular ;  and  D  a  point  farther 
distant  from  C  than  A  and  B. 

1st.  Let  all  the  points  A,  C,  B,  and  D 
be  in  the  same  straight  line.  Suppose  a 
plane  passed  through  them  and  the  point  P ;  then  will  all  the 
lines  PA,  PC,  PB,  and  PD  be  in  the  plane  PAD;  and  AD,  the 
line  of  intersection  of  the  two  planes,  will  be  a  base  line  for  the 
figure  A  DP.  Since  PC  is  perpendicular  to  the  plane  MN,  it  is 
perpendicular  to  AD.  (Del.  1.)  PA  and  PB  are  oblique  lines 
drawn  from  P  to  points  equally  distant  from  the  foot  of  PC ; 
hence,  PA  =  PB.  (Bk.  2,  Th.  6,  part  2.)  And  as  D  is  farther 
distant  from  C,  PD  is  longer  than  PA  or  PB.  (Part  3,  same 
Prop.) 

2d.  Other  oblique  lines,  as  PE,  PF,  and  PH,  may  be  drawn  to 
different  points  of  M  N,  at  the  same  distance  from  C  as  A  and  B, 
and  the  planes  PCE,  PC F,  and  PCH  passed;  the  triangles  PCE, 
PCF,  and  PCH  will  be  right-angled  triangles,  equal  to  PCA  and 
PCD  ;  hence,  PE  =  PF  =  PA  =  PB. 

Rem.— If  a  circumference  be  drawn  with  C  as  a  center  and  a 
radius  equal  to  CA,  it  will  pass  through  all  the  points  A,  E,  F,  B, 
and  H,  and  the  point  D  will  be  without  the  circumference. 


BOOK     VI. 


65 


THEOREM    II. 

A  line  ivhich  is  perpendicular  to  two  lines  of  a  plane, 
at  their  intersection,  is  perpendicular  to  any  other  line  of 
the  plane  passing  through  this  intersection  and  therefore 
perpendicular  to  the  plane. 

Let  PC  be  perpendicular  to  AB  and 
DE  at  C,  the  point  of  intersection  of  the 
two  lines  in  the  plane  MN.  With  C  as  a 
center  and  any  radius  describe  a  circum- 
ference cutting  the  two  lines  in  A,  B,  D, 
and  E,  and  draw  PA,  PB,  PD,  and  PE. 
Through  C  draw  any  other  line  as  FG, 
terminating  in  the  circumference.  PC 
will  also  be  perpendicular  to  FG  ;  as  F  and 

G  are  equally  distant  from  C,  PF  is  equal  to  PG  ;  hence  the  line 
PC  has  two  points  P  and  C  equally  distant  from  F  and  G,  the 
extremities  of  the  line  FG  ;  hence  PC  is  perpendicular  to  FG,  any 
line  of  the  plane  MN,  and  is  therefore  perpendicular  to  every  line 
in  the  plane;  consequently  perpendicular  to  the  plane.    (Def.  1.) 

Cor. — The  two  sides  of  an  angle,  two  parallel  lines,  or  three 
points  not  in  the  same  straight  line,  determine  the  position  of  a 
plane. 

THEOREM    III. 

If  from  the  foot  of  a  perpendicular  to  a,  plane  a  line  be 
drawn  at  right  angles  to  any  line  of  that  plane f  and  the 
point  of  intersection  of  these  two  lines  be  joined  with  any 
point  in  the  perpendicular,  the  last  line  will  be  perpendic- 
ular to  the  line  in  the  plane  to  which  the  line  was  drawn 
at  right  angles. 

From  P,  the  foot  of  the  perpendicular 
AP,  draw  PB  perpendicular  to  HY,  any  line 
in  the  plane  MN.  Make  BC  and  BD  equal, 
and  draw  PC  and  PD,  which  will  also  be 
equal;  and  then  draw  AC  and  AD,  which 
will  also  be  equal.  (Bk.  2,  Prob.  6,  part  2.) 
And  since  AB  has  two  points,  A  and  B, 
equally  distant  from  D  and  C,  AB  is  perpen- 
dicular to  DC.     (Bk.  1,  Th.  9.) 


p 

i       A' 

p 

\  K    Y 

M 


ELEMENTS     OF     GEOMETRY. 


A 

N 

wi/ 

1 

/x 

/ 

J 

/ 

/B 

Q 

J 

/ 

Cor.  1. — Since  BA  is  perpendicular  to  DC,  when  drawn  from  any  point 
in  the  line  FA  or  PA  produced  indefinitely ;  and  as  in  every  position  BA  lies 
in  the  plane  A?B,  it  is  also  perpendicular  to  DC  when  it  becomes  parallel  to 
PA  and  is  also  perpendicular  to  PB,  and  hence  perpendicular  to  the  plane 
MN;  consequently,  if  one  of  two  parallels  is  perpendicular  to  a  plane,  the 
)ther  is  also  perpendicular  to  the  same  plane. 

COB,  2. — Two  lines  perpendicular  to  the  same  plane  are  parallel. 

THEOEEM    IV. 

If  two  parallel  planes  are  intersected  by  a  third  plane, 
the  lines  of  intersection  will  be  parallel. 

Let  the  parallel  planes  MN  and  PQ  be 
intersected  by  the  plane  ABCD;  then  will 
the  lines  of  intersection  AB  and  CD  be 
parallel. 

Make  AB  and  CB  of  the  same  length, 
and  draw  AC  and  BD ;  the  line  AB  lies  in 
the  plane  MN,  and  CD  in  the  plane  PQ; 
as  the  planes  are  parallel,  they  will  never 
D  meet;  hence,  AB  and  CD  will  never  meet; 

but  AB  and  CD  also  lie  in  the  same  plane  ABCD,  and  are  there- 
fore parallel. 

Cor.  1. — The  figure  ABCD  is  a  parallelogram. 

Cor.  2. — Parallel  lines  intercepted  between  parallel  planes  are  parallel. 
Rem. — A  plane  may  always  be  made  to  pass  through  parallel  lines,  as 
all  parallels  have  the  same  direction. 

THEOREM    V. 

If  in  parallel  planes,  angles  are  formed  whose  sides 
respectively  take  the  same  direction,  the  angles  will  be 
equal. 

In  the  planes  M  and  N,  let  A  and  B  be 
angles  whose  sides  respectively  take  the 
same  direction.  Since  the  sides  of  tile 
angles  respectively  take  the  same  direction, 
if  the  one  angle  is  placed  on  the  other, 
they  will  coincide,  and  consequently  are 
equal. 

Cor.  1, — If  the  vertices  A  and  B  are  joined 
by  a  straight  line,  and  planes  passed  through  this 
line  and  the  corresponding  sides  of  the  two  angles,  a  diedral  angle  will  be 
formed. 

Cor.  2. — If  a  diedral  angle  is  cut  by  parallel  planes,  the  plane  angles 
formed  are  equal. 


M 

A 

■/\ 

i 

N 

3 

j 

BOOK     V  I  I 


DEFINITIONS. 

1.  A  Prism  is  a  solid,  two  of  whose  faces  are  equal  parallel 
polygons,  which  are  .termed  the  lower  and  upper  bases,  whilst 
the  other  faces  are  parallelograms  which  form  the  convex  surface. 

Rem. — The  bases  may  be  polygons  of  any  number  of  sides. 

2.  A  Right  Prism  has  its  lateral  edges  perpendicular  to 

its  bases. 

3.  An  Oblique  Prism  has  its  lateral  edges  oblique  to 
its  bases. 

4.  The  Altitude  of  a  prism  is  the  perpendicular  distance 
between  its  bases. 

5.  A  Regular  Prism  is  a  right  prism  having  regular 
polygons  for  its  bases. 

6.  A  Parallelopipedon  is  a  prism  having  its  bases  and 
its  faces  parallelograms,  the  opposite  faces  necessarily  equal. 

7.  If  the  bases  and  faces  are  rectangles,  it  is  called  a 
Rectangular  Parallelopipedon. 

8.  If  they* are  all  equal  squares,  it  is  a  Cube. 

Rem. — A  solid  is  said  to  have  three  dimensions,  to  which 
special  attention  must  be  given  ;  thus,  the  dimensions  of  a  rect- 
angular parallelopipedon  are  the  length  and  the  perpendicular 
breadth  of  the  base,  regarding  it  as  a  plane  figure,  and  the  per- 
pendicular distance  between  the  two  bases. 

In  a  prism  of  any  shape,  the  dimensions  of  the  base  are  the 
same  as  those  of  a  plane  figure,  and  the  altitude  is  the  perpen- 
dicular distance  between  the  bases. 


ELEMENTS     OF     GEOMETEY. 


THEOREM    I. 

TJie  sections  formed  by  parallel  -planes  cutting  a  prism, 
cure  equal  polygons. 

Let  abcde  and  a'b'c'd'e'  be  sections  formed  by- 
parallel  planes  cutting  the  prism  ;  ab  and  a'b'  are 
parallel,  being  the  intersection  of  parallel  planes 
with  a  third  plane  (Book  6,  Th.  4) ;  and  they  are 
equal,  as  they  are  parallels  between  two  other 
parallels;  so  also  for  the  same  reasons  are  be  = 
b'c',  cd  =  c'd',  etc.  ;  and  since  the  angles  are 
formed  by  parallel  planes  cutting  diedral  angles, 
they  are  respectively  equal  (Bk.  6,  Th.  5,  Cor.  2) ; 
hence,  the  sections  are  equal  polygons. 

Cor. — If  the  sections  are  parallel  to  the  bases, 
then  will  they  be  equal  polygons. 


e' 

Y 

3 

"2j 

e\ 

^ 

h 

fi\ 

*<*? 

-i 

A 

a 

b 

THEOREM    II. 

Tlie  lateral  surface  of  a  right  prism  is  equal  to  the 
product  of  its  perimeter  and  altitude. 

As  each  face  is  a  rectangle,  its  surface  is  equal  to  the  product 
of  its  base  and  altitude ;  hence,  the  lateral  surface  of  the  prism 
is  the  product  of  the  sum  of  the  sides  of  the  base  and  the  common 
altitude;  but  the  sum  of  the  sides  of  the  base  is  the  perimeter; 
hence,  the  lateral  surface  of  a  prism  is  the  product  of  its  perim- 
eter and  altitude. 

Cor.  1. — If  the  prism  be  oblique,  its  faces  will  be  parallelo- 
grams instead  of  rectangles,  and  is  measured  accordingly. 

Cor.  2. — The  convex  surface  of  a  cylinder  is  equal  to  the 
product  of  the  circumference  of  its  base  and  altitude.  For  a 
prism  having  a  regular  polygon  for  its  base  becomes  a  cylinder, 
by  increasing  the  number  of  sides  indefinitely,  and  the  perimeter 
of  the  polygon,  forming  its  base,  becomes  the  circumference  of 
the  base  of  the  cylinder. 

Cor.  3.  Convex  surface  of  a  cylinder  =  4rrR2, 
Surface  of  the  two  bases  =  2ttFP, 
Entire  surface  of  the  cylinder  =  67rR2. 


BOOK     VII.  69 


THEOREM    III. 

The  volume  of  a  rectangular  parallelopipedon  is 
equal  to  the  product  of  its  three  dimensions. 

The  area  of  the  rectangular  base  is  the  product  of  its  two 
dimensions,  and  for  every  unit  in  altitude  there  will  be  as  many 
solid  units  as  there  are  square  units  in  the  base;  hence,  the 
volume  is  equal  to  the  product  of  the  three  dimensions. 

Cor.  1. — A  rectangular  parallelopipedon  can  be  divided  into 
two  equal  triangular  prisms ;  and  the  volume  of  each  is  equal  to 
the  area  of  the  base  multiplied  by  the  altitude. 

Cor.  2. — The  volume  of  any  right  prism  is  equal  to  the 
area  of  its  base  multiplied  by  its  altitude;  as  any  prism  may  be 
divided  into  triangular  prisms. 

Cor.  3. — A  right  prism  having  for  its  base  a  regular  polygon, 
if  the  number  of  sides  of  the  polygon  be  indefinitely  increased,  it 
becomes  a  cylinder ;  hence  the  volume  of  a  cylinder  is  equal  to 
the  area  of  the  base  multiplied  by  its  altitude. 

Cor.  4. — An  oblique  prism  is  measured  in  the  same  way  ;  as 
the  area  of  a  rectangle  and  a  parallelogram  of  the  same  dimen- 
sions is  equal ;  so  also  is  the  volume  of  the  oblique  prism  equal 
to  that  of  the  right  prism  of  the  same  dimensions. 

Scho.  1. — The  above  results  are  true  if  one  or  more  of  the 
dimensions  are  fractional,  as  the  common  denominator  of  the 
fractions  will  be  the  denomination  of  the  unit  of  measure. 

Scho.  2.— If  the  dimensions  are  incommensurable,  the  unit  of 
measure  will  be  an  infinitesimal. 


70 


ELEMENTS     OF     GEOMETRY, 


THEOREM    IV. 

A  triangular  prism  may  be  divided  into  three  equiv- 
alent pyramids. 

1st.  Let  ABC  DEF  be  a  triangular  prism.  Pass 
a  plane  through  the  three  points,  A,  B,  and  F, 
cutting  off  the  pyramid  ABC-F,  having  the  tri- 
angle ABC  for  its  base,  and  the  altitude  of  the 
prism  for  its  altitude. 

2d.  Pass  a  plane  through  BFD,  cutting  off  the 

pyramid  DEF-B,  the  upper  base  for  its  base,  and 

having  the  altitude  of  the  prism  for  its  altitude ;  hence  it  has  the 

same   dimensions   as   the   first  pyramid,   and    is    consequently 

equal  to  it. 

3d.  Take  away  the  first  pyramid  and  place  the  two  remaining 
upon  ABDE  as  a  base;  their  vertices  will  be  in  F,  and  hence  will 
have  the  same  altitude  ;  their  bases  are  equal,  as  each  is  the  half 
of  the  parallelogram  ABED,  as  the  diagonal  BD  divides  it  into 
equal  parts. 

And  as  the  first  and  second  are  equal,  and  also  the  second 
and  third,  it  follows  that  the  first  and  third  are  equal ;  hence 
they  are  all  equal. 

Cor. — The  volume  of  a  triangular  pyramid  is  equal  to  the 
area  of  its  base,  multiplied  by  one-third  its  altitude. 


THEOREM    V. 

The  volume  of  the  frustum  of  a  triangular  pyramid 
is  equivalent  to  that  of  three  pyramids,  two  of  which 
shall  have  for  their  altitudes  the  altitude  of  the  frustum, 
and  for  their  bases  respectively  the  lower  and  zipper 
bases  of  the  frustum,  and  the  third  pyramid  shall  be 
a  mean  proportional  between  the  other  two. 

Let  ABC  DEF  be  the  frustum  of  a  pyramid. 

1st.  Pass  a  plane  through  ABF,  cutting  off 
a  pyramid,  having  the  lower  base  of  the  frus- 
tum for  its  base,  and  the  altitude  of  the  frustum 


for  its  altitude. 


Designate  it  P. 


BOOK     VII.  7i 

2d.  Draw  the  diagonal  DB  and  pass  a  plane  through  DBF, 
cutting  off  a  pyramid,  having  for  its  base  the  upper  base  of  the 
frustum,  and  for  its  altitude  the  altitude  of  the  frustum.  Desig- 
nate this  pyramid  P'. 

Take  away  the  pyramid  P  and  place  the  remnant  on  ABED 
for  its  base,  and  being  cut  by  the  plane  DBF,  forms  two  pyra- 
mids, having  their  vertices  in  F;  hence  they  have  the  same 
altitude;  the  one  P'  has  DEB  for  its  base;  and  the  third,  which 
designate^,  has  for  its  base  ABD. 

Since  P  and  P'  have  the  same  altitudes,  they  are  in  propor- 
tion to  their  bases;  and  as  their  bases  are  similar  triangles,  which 
are  proportional  to  the  squares  of  their  homologous  sides,  AB 
and  DE  are  homologous  sides;  let 

S  =  AB, 

and  s  —  DE, 

then  P  :  P'    ::    S2  :  s2;  (1) 

p  and  P'  have  the  same  altitude ;  hence  they  are  in  proportion  to 
the  areas  of  their  bases,  ABD  and  DEB,  which  triangles  have  the 
same  altitude,  and  are  therefore  proportional  to  their  bases, 
S  and  5 ;  hence, 

p  :   P' ■::  S:  s;  (2) 

squaring  the  proportion, 

f  :  P2    ::    S2  :  s2 ;  (3) 

.-.  (1)  and  (3)  have  an  equal  ratio ;  hence, 

P  :  P'    ::    p2  :  P'2. 

Multiplying  the  extremes  and  means, 

P  x  P'2  =  P'  x  p2; 

and  dividing  by  P',        P  x  P'  =  p2; 

that  is,  the  third  pyramid  is  a  mean  proportional  between  the 
other  two. 


72  ELEMENTS     OF     GEOMETRY 


THEOREM    VI. 

The  convex  surface  of  a  right  pyramid  is  equal  to  the 
product  of  the  perimeter  of  its  base  and  one-half  its 
slant  height. 

The  base  of  the  right  pyramid  ABCDE  is  a  reg- 
ular polygon.  A  perpendicular  from  the  vertex  S 
would  fall  upon  the  center  of  the  polygon.  The 
faces  will  all  be  equal  isosceles  triangles;  and  the 
surface  of  each  is  the  product  of  its  base  and  one- 
half  its  altitude ;  the  altitude  of  each  triangle  is 
the  slant  height  of  the  pyramid ;  consequently, 
the  entire  convex  surface  is  the  perimeter  of  the 
base  multiplied  by  one-half  the  slant  height. 

Cor.  1. — If  the  number  of  sides  of  the  regular  polygon  of  the 
base  be  indefinitely  increased,  the  polygon  becomes  a  circle,  and 
the  pyramid  a  cone ;  hence  the  convex  surface  of  a  cone  is  one- 
half  the  product  of  the  circumference  of  the  base  and  the  slant 
height. 

Cor.  2. — If  a  plane  dbcde  cut  the  pyramid  or  cone  parallel  to 
the  base,  then  the  portion  between  the  parallel  bases  will  be  a 
frustum  of  a  pyramid  or  cone.  The  sides  of  the  frustum  are  all 
trapezoids,  each  of  which  is  measured  by  the  product  of  one-half 
the  sum  of  the  parallel  bases  and  altitude  ;  hence,  the  convex 
surface  of  the  entire  frustum  is  one-half  the  product  of  the  sum 
of  the  two  perimeters  and  the  slant  height. 

Cor.  3. — The  volume  of  the  frustum  of  a  cone  is  equal  to  that 
of  three  cones,  two  having  for  their  altitude  the  altitude  of  the 
frustum,  and  for  their  bases  respectively  the  lower  and  upper 
bases  of  the  frustum,  and  the  third  cone  a  mean  proportional 
between  the  other  two. 


BOOK      VIII 


DEFINITIONS. 

1.  A  Polyedral  Angle  is  the  divergence  of  three  or  more 
planes  from  the  point  formed  by  their  common  intersection. 

2.  An  angle  formed  by  three  planes  is  called  a  Triedral 
Angle. 

3.  The  plane  angles  which  form  a  triedral  angle  are  called 
Facial  Angles. 

4.  A  Sphere  is  a  solid  every  point  of  whose  surface  is 
equally  distant  from  a  point  within  called  the  centre. 

5.  The  distance  from  the  center  to    the  surface    is    the 
Radius  of  the  sphere. 


THEOREM    I. 

The  sum  of  any  two  of  the  plane  angles  which  form  a 
triedral  angle  is  greater  than  the  third  angle. 

Let  the  plane  angle  ASB  be  greater  than 
either  of  the  other  angles  which  form  the 
triedral  angle  S. 

In  the  plane  ASB  draw  AB,  making 
SA  =  SB,  and  on  the  plane  angle  ASB  place 
the  angle  BSC;  SB  on  SB,  and  SC  will 
take  the  direction  of  SD,  and  make  SC  =  SD; 
then  as  AB  <  AC  +  BC,  and  BC  =  BD,  and 
taking  BC  =  BD  from  each  side,  AD  <  AC. 

The  two  triangles  ASC  and  ASD  have  two  sides  respectively 
equal  and  the  third  side  AC  >  AD  ;  hence  the  angle  ASC  >  ASD ; 
therefore  the  sum  of  the  two  plane  angles  BSC  and  ASC  is 
greater  than  the  third  angle  ASB. 
4 


n 


ELEMENTS     OF     GEOMETRY 


THEOREM    II. 


The  sum  of  all  the  angles  which  form  a  polyedral 
angle  is  less  than  four  right  angles. 

Let  S  be  the  vertex  of  a  polyedral 
angle,  and  pass  a  plane  cutting  the  planes 
and  forming  the  polygon  ABCDE. 

From  any  point  as  0  within  the  poly- 
gon, draw  lines  to  the  extremities  of  all  its 
sides  ;  there  will  be  as  many  triangles  as 
faces  forming  the  polyedral  angle. 

At  each  vertex  of  the  polygon  there  is 
a  triedral  angle,  formed  by  one  plane  angle 
of  the  polygon  and  two  in  the  faces  of  the 
polyedral  angle  S;  the  one  angle  in  the  polygon  is  less  than  the 
sum  of  the  two  others  (Th.  1);  but  as  the  number  of  the  tri- 
angles in  the.  polygon  is  the  same  as  of  the  plane  angles  forming 
the  polyedral  angle  S,  hence  the  sum  of  all  the  angles  of  the  tri- 
angles in  the  polygon  is  equal  to  the  sum  of  all  the  angles  of  the 
triangles  forming  the  polyedral  angle  S. 

And  as  the  sums  of  all  the  angles  of  the  polygon  are  less  than 
the  sums  of  all  the  angles  on  the  faces  at  the  points  A,  B,  C,  etc., 
at  which  triedral  angles  are  formed,  on  the  faces  there  are  two 
angles  at  each  vertex,  whilst  in  the  polygon  there  is  but  one,  the 
remaining  angles  of  the  triangles  of  the  polygon  must  be  greater 
than  the  sum  of  the  angles  forming  the  polyedral  angle  at  S.  The 
sum  of  all  the  angles  at  0  is  four  right  angles;  hence  the  sum  of 
all  the  plane  angles  forming  the  polyedral  angle  S  is  less  than 
four  right  angles. 


BOOK*    VIII. 


75 


THEOREM    III. 

The  surface  generated  by  the  revolution  of  a  regular 
semi-polygon  about  the  diameter  of  the  circumscribed 
circle  as  an  axis,  is  equal  to  the  circumference  of  the 
inscribed  circle  multiplied  by  this  axis. 

Let  ABCD  be  one-half  of  a  regular  polygon, 
which  being  revolved  about  AD  the  diameter  of 
the  circumscribed  circle,  as  an  axis,  the  sur- 
face generated  by  the  perimeter  is  2ixO«x  AD. 

The  triangles  AB&  and  CDc  will  generate 
equal  cones,  and  the  rectangle  BbCc  will  gen- 
erate a  cylinder. 

The  surface  generated  by  AB  =  2/txex AB, 
"  BC  =  2n  x  nO  x  BC, 
"        "  "         "  CD  =  2^x«'m'xCD. 


The  triangles  Oam  and  AB#  are  similar;  consequently, 

AB  :Oa  ::   1Kb  :  am\    .'.     ABxam  =  Oaxfcb\ 
hence  also,  a'm  x  CC  =  Oa'  x  cD. 

Surface  generated  by  AB  =  2n  x  Oa  x  kb, 

"  "         «   BC  =  2nxOnxbc, 

"   CD  =  2nXOa'xcD. 

Oa  —  On  =  Oa',  each  equal  to  the  radius  of  the  inscribed  circle. 
By  addition,        Whole  surface  ==  2<i  xOax  AD. 

Ook.  1. — When  the  number  of  the  sides  of  the  semi-polygon 
is  indefinitely  increased,  it  becomes  a  semicircle ;  the  radius  of 
the  inscribed  circle  becomes  equal  to  that  of  the  circumscribed, 
and  the  figure  generated  a  sphere ;  and  its  surface  ==  2n  x  r  x 
2r  =  4.nr2 ;  that  is, 

The  surface  of  a  sphere  is  equal  to  the  circumference  of  a 
great  circle  multiplied  by  the  diameter. 

Cor.  2. — The  surfaces  of  spheres  are  to  each  other  as  the 
squares  of  their  radii ;  for,  let  R  and  R'  be  the  radii  of  two 
spheres,  their  surfaces  will  be  4?iR2  and  4ttR'2;  hence, 

4^R2  :  4ttR'2   ::    R2  :  R'2. 


76  ELEMENTS     OF     GEOMETEY. 


THEOREM     IV. 

The  volume  of  a  sphere  is  equal  to  the  product  of  its 
surface  and  one-third  of  its  radius. 

Take  a  cube,  each  side  say  2  inches,  and  suppose  a  sphere 
inscribed.  Consider  each  face  as  the  base  of  a  pyramid  whose 
vertex  is  in  the  center  of  the  inscribed  sphere,  whose  radius  is 
one  inch,  which  is  also  the  altitude  of  each  pyramid. 

The  volume  of  each  pyramid  is  equal  to  the  product  of  its 
base,  and  one-third  of  its  altitude  ;  and,  the  volume  of  all  the 
pyramids,  which  equals  that  of  the  cube,  is  the  whole  surface  of 
the  cube  multiplied  by  one-third  the  radius  of  the  inscribed 
sphere. 

The  cube  has  eight  triedral  angles ;  let  each  of  these  be  cut 
by  a  plane  tangent  to  the  inscribed  sphere,  and  perpendicular 
to  a  straight  line  drawn  from  the  center  of  the  sphere  to  the 
vertex  of  each  triedral  angle  ;  thus,  a  new  set  of  pyramids  will 
be  formed,  each  having  the  same  altitude  as  the  former ;  the 
number  of  bases  will  be  increased,  and  the  sum  of  all  the  bases 
will  be  the  surface  of  the  solid  figure  remaining. 

Continue  to  pass  planes  indefinitely,  tangents  to  the  inscribed 
sphere,  until  the  surface  of  the  figure  becomes  the  surface  of  a 
sphere,  and  its  volume  will  be  the  surface  of  the  sphere  multi- 
plied by  one-third  of  the  radius  ;  that  is, 

Volume  of  sphere     =  4"R2x-JR  =  $"R3; 

Volume  of  cylinder  =  ?iR2x2R  =  2nR3; 

hence,    Sur.  of  sphere  :  Sur.  of  cylinder  : :  4:6  : :  2:3. 

Vol.  of  sphere  :  Vol.  of  cylinder  ::  $  :  2  ::  4:6   ::   2:3; 

.*.    Sur.  sphere  :  sur.  cylinder  : :  vol.  sphere  :  vol.  cylinder. 

Rem. — When  a  plane  touches  a  sphere  at  but  one  point,  it  is 
tangent  to  the  sphere,  and  the  plane  is  perpendicular  to  the 
radius  drawn  to  this  point. 

V 


4C  A4utC^  fiMhJ^CL  V  J   hJs 


BOOK    VIII.  77 

THEOREM    V. 
Every  section  of  a  sphere  made  by  a  plane  is  a  circle. 

Let  ACBD  be  a  section  of  a  sphere  whose 
center  is  0  ;  draw  OE  perpendicular  to  the  sec- 
tion ;  OA,  OB,  OC,  and  OD,and  any  other  line 
from  0  to  the  intersection  of  the  plane  and 
the  surface  of  the  sphere,  will  all  be  equal,  as 
they  are  radii  of  the  sphere ;  and  as  OE  is 
perpendicular  to  the  section,  it  will  pass 
through  the  middle  point  of  AB  or  any  other  line  passing 
through  E  and  terminating  in  the  surface  of  the  sphere ;  hence, 
ACBD  is  a  circle  and  E  is  its  center. 

Rem. — If  the  plane  pass  through  the  center  of  the  sphere, 
the  circle  formed  will  be  a  great  circle  ;  if  it  do  not  pass  through 
the  center  it  will  form  a  small  circle. 

Cor.  1. — A  line  drawn  from  the  center  of  a  sphere  perpen- 
dicular to  a  small  circle  passes  through  its  center,  or  a  line 
perpendicular  to  a  small  circle  at  its  center  passes  through  the 
center  of  the  sphere,  and  its  extremities  are  poles  of  the  small 
circle  and  of  every  circle  whose  plane  is  parallel  to  that  of  the 
small  circle. 

Cor.  2. — The  pole  of  a  circle  is  equally  distant  from  every 
point  in  the  circumference;  in  the  case  of  the  great  circle  both 
poles  are  equally  distant,  each  being  a  quadrant's  distant ;  in  the 
case  of  the  small  circle,  the  one  is  greater  the  other  less  distant 
than  a  quadrant. 

Cor.  3. — The  farther  from  the  center  the  less  the  circle. 


THEOREM    VI. 

A  great  circle  divides  the  sphere  into  two  equal  parts. 

The  plane  of  the  great  circle  passes  through  the  center  of  the 
sphere,  and  divides  it  into  two  parts,  each  of  which  lias  the 
great  circle  for  its  base,  and  every  point  of  the  convex  surface  of 
each  part  is  equally  distant  from  the  center  of  their  common  bases ; 
hence,  the  two  parts  must  coincide  and  consequently  are  equal. 

Rem. — Each  part  is  called  a  hemisphere. 


78  ELEMENTS     OF     GEOMETRY. 


THEOREM    VII. 

.  The  intersection  of  two  great  circles  is  a  diameter  of 
the  sphere. 

The  intersection  of  two  planes  is  a  straight  line,  and  as  each 
plane  of  a  great  circle  passes  through  the  center  of  the  sphere, 
the  center  is  one  point  of  their  intersection ;  hence  their  inter- 
section is  a  straight  line  passing  through  the  center  of  the  sphere. 
This  straight  line  is  a  diameter. 

Cor. — The  intersection  of  the  circumferences  on  the  surface 
of  the  sphere  will  be  the  extremities  of  the  diameter,  180°  distant ; 
hence  they  bisect  each  other. 

Def.  1. — The  portion  of  the  surface  of  a  sphere  included  be- 
tween two  semi-circumferences  of  great  circles  is  called  a  Lime. 

Def.  2.— A  Spherical  Polygon  is  a  portion  of  the  sur- 
face of  a  sphere  bounded  by  three  or  more  arcs  of  great  circles. 
The  arcs  form  the  sides  of  the  polygon,  each  of  which  is  less  than 
a  semi-circumference. 

Rem. — The  sides  of  a  spherical  polygon  correspond  to  the 
facial  angles  of  the  polyedral  made  at  the  center  of  the  sphere  by 
the  planes  of  the  sides  of  the  polygon,  and  the  angles  of  the 
polygon  correspond  to  the  diedrals  of  the  same  polyedral  angle. 


THEOREM    VIII. 

The  shortest  distance  between  any  tivo  points  on  the 
surface  of  a  sphere  is  traced  on  the  arc  of  a  great  circle. 

The  truth  of  this  theorem  is  evident  from  Th.  8,  Bk.  5.  For, 
when  two  unequal  circumferences  intersect  on  the  surface  of  a 
sphere,  the  intercepted  arcs  hold  the  same  position  in  regard  to 
each  other;  as,  when  a  large  and  a  smaller  circle  intersect  on  the 
same  plane  ;  from  which  it  is  evident  that  the  intercepted  arc  of 
the  greater  circle  is  less  than  that  of  the  smaller ;  and,  as  on  a 
sphere  the  circumference  of  a  great  circle  is  the  largest  that  can 
be  described  upon  it,  hence  the  shortest  distance  between  any 
two  points  on  the  surface  of  a  sphere  must  be  traced  on  it. 


BOOK     VIII, 


79 


PROBLEM    I. 

To  pass  a  small  circle  through  any  three  points  on  the 
surface  of  a  sphere,  not  in  the  circumference  of  a  great 
circle. 

Let  A,  B,  and  C  be  the  three  points. 
Join  A  and  C,  and  B  and  C  by  arcs  of 
great  circles.  Pass  arcs  of  great  circles 
perpendicularly  through  the  middle  points 
of  AC  and  CB,  as  in  Plane  Geometry,  and 
their  intersection  0  will  be  the  pole  of  a 
small  circle,  from  which,  with  the  distance 
between  the  points  of  the  dividers  equal  to 
OA,  describe  the  small  circle,  which  will  pass  through  A,  C,  and  B. 


PROBLEM    II 

To  pass  a  great  circle  through  any  two  points  on  the 
surface  of  a  sphere. 

Let  A  and  B  be  any  two  points  on  the 
surface  of  a  sphere.  Make  either  point,  as  A, 
a  pole,  and  from  it  describe  a  circumference 
of  either  a  small  or  a  great  circle ;  and  from 
the  other  point,  as  B,  pass  the  arc  of  a  great 
circle  cutting  the  circumference  CD  at  right 
angles;  it  will  pass  through  the  point  A.  The 
pole  of  the  great  circle  must  be  taken  at  a  quadrant's  distance 
from  B  and  from  E. 

Rem. — The  point  on  the  circumference  of  the  circle  through 
which  the  arc  of  a  great  circle  passes,  is  found  by  Prob.  2,  Book  I, 
and  from  the  pole  of  this  point  and  B,  a  quadrant's  distance  from 
each,  the  arc  is  drawn. 

THEOREM    IX. 

In  every  spherical  triangle,  the  sum  of  any  two  sides 
is  greater  than  the  third  side. 

Let  ABC  be  a  spherical  triangle ;  then 
will  AC  +  BC>  A  B.  Join  the  vertices.  A,  B, 
and  C,  with  0,  the  center  of  the  sphere,  and 
a  triedral  angle  is  formed,  the  arcs  of  whose 
facial  angles  are  the  sides  of  the  spherical 
triangle;  but  (Th.  1)  "the  sum  of  any  two 
of  the  plane  angles  which  form  a  triedral 


80 


ELEMENTS     OF     GEOMETRY 


angle  is  greater  than  a  third ;  hence,  the  sum  of  any  two  sides  of 
a  spherical  triangle  is  greater  than  the  third  side. 


THEOREM    X. 

TJie  sum  of  the  three  sides  of  a  spherical  triangle  is 
less  than  the  circumference  of  a  great  circle. 

Let  ABC  be  a  spherical  triangle.     Pro- 
duce AB  and  AC  until  they  meet  in  D. 
The  arcs  ABD  and  ACD  are  semi-circum- 
ferences, since  two  great  circles  always 
bisect  each  other. 
In  the  triangle  BCD,  the  sum  of  the  two  sides  CD  and  BD  is 
greater  than  the  third  side  BC  ;  hence  the  sum  of  the  three  sides 
AB,  AC,  and  BC  is  less  than  the  circumference  of  a  great  circle. 

Rem. — The  sides  of  a  spherical  triangle  are  arcs  which  cor- 
respond with  and  measure  the  facial  angles  of  a  triedral  angle, 
the  vertex  of  which  is  at  the  center  of  the  sphere. 


THEOREM    XI. 

The  sum  of  all  the  sides  of  a  spherical  polygon  is  less 
than  the  circumference  of  a  great  circle. 

Let  ABCDE    be    a  spherical    polygon. 

Produce  AB  and  DC  until  they  meet  in  G  ; 

also  produce  AE  and  CD  until  they  meet  in 

F.     BC  is  less  than   BG  +  CG,  and  DE  is 

less  than  EF-f-FD;   hence  the  sum  of  the 

sides  of  the  polygon  is  less  than  the  sum  of 

the  sides  of  the  triangle  AFG  ;  and  the  sum 

of  the  sides  of  the  triangle  is  less  than  the  circumference  of  a 

great  circle  (Th.  10) ;  much  more  then  is  the  sum  of  all  the  sides 

of  the  polygon  less  than  the  circumference  of  a  great  circle. 


BOOK     VIII 


81 


THEOKEM    XII. 

If  from  the  vertices  of  the  angles  of  a  spherical  tri- 
angle as  poles,  with  a  distance  between  the  points  of  the 
dividers  eqaal  to  a  quadrant,  arcs  be  drawn  forming 
another  spherical  triangle,  the  vertices  of  this  triangle 
will  be  respectively  the  poles  of  the  sides  of  the  first 
triangle. 

Let  ABC   be  a  spherical  triangle,  and 
then  with  each  vertex  as  a  center  and  a 
distance  between  the  points  of  the  dividers 
equal  to  ninety  degrees,  describe  the  polar 
triangle    DEF.      First  with   A  as  a    pole 
describe   the  arc   EF,    with    B  as  a    pole 
describe     DF,    and    with     C     as    a     pole 
describe    DE;    in   each    case    the    distance 
between  the  points  of   the   dividers   being   90   degrees.     Since 
AE  =:  90°  and  CE  =  90°,  E  is  the  pole  of  AC  ;  and  since  BD  = 
90°  and  DC  =  90°,  D  is  the  pole  of  BC  ;  and  as  BF  =  90°  and 
AF  =  90°,  F  is  the  pole  of  AB. 


THEOREM    XIII. 

Any  angle  in  one  of  two  polar  triangles  is  measured 
by  a  semi- circumference  minus  the  side  opposite  of  the 
other  triangle. 

Let  ABC  and  DEF  be  triangles  polar  to 
each  other  ;  produce  the  sides  of  ABC  until 
they  meet  those  of  DEF;  A  is  the  pole  of 
the  arc  GH  by  which  the  angle  A  is  meas- 
ured, E  is  the  pole  of  KH,  and  F  is  the  pole 
of  LG;  hence,  EH  =  90°  and  FG  ==  90°; 
hence,  GH  =  180°— EF;  that  is,  the  angle 
A  is  measured  by  a  semi-circumference 
minus  the  side  opposite  in  its  polar  triangle ;  so  also  in  regard  to 
each  of  the  other  angles. 

Cor.  to  10th  and  13th  Th. — The  sum  of  the  three  angles  of  a 
spherical  triangle  is  less  than  six  right  angles  and  greater  than  two. 


82 


ELEMENTS     OF     GEOMETRY. 


Cor.  2. — A  spherical  triangle  ma)7  have  two  or  even  three 
right  angles,  or  as  many  obtuse  angles  ;  when  it  has  three 
right  angles  it  is  called  the  trirectangular  triangle,  whose  surface 
is  equal  to  one-eighth  the  surface  of  the  sphere. 

Cor.  3. — The  sum  of  the  three  angles  of  a  spherical  triangle 
is  not  a  constant  quantity,  but  varies  between  two  and  six  right 
angles,  never  reaching  either  of  these  limits. 

Rem. — The  excess  of  the  sum  of  the  angles  of  a  spherical 
triangle  over  two  right  angles  is  called  the  spherical  excess. 


THEOREM    XIV. 

A  lune  is  to  the  surface  of  a  sphere  as  the  are  lohich 
measures  its  angle  is  to  the  circumference  of  a  great 
circle. 

Surface,  of  lune  :  8  trirectangular  tri.   : :   angle  A  :  4 ; 
.-.    4L  =  Ax8T,    and     L  =  A  x  2T. 
Cor. — The  surface  of  a  lune  is  equal  to  its  angle 
multiplied  by  twice  the  trirectangular  triangle. 

Rem. — As  the  right  angle  is  the  unit,  the  angle  will 
be  indicated  by  a  fraction  ;  as  fjj-,  which  is  43  degrees. 


THEOREM    XV. 

The  area  of  a,  spherical  triangle  is  equal  to  its  spher- 
ical excess  multiplied  by  the  trirectangular  triangle. 

Let  ABC  be  the  spherical  triangle. 

The  triangles  ADE  and  AFG  form  a  lune 
?=  angle  A  x  2T; 

the  triangles  BGH  and  BID  form  a  lune 
=  angle  B  x  2T; 

the  triangles  CFI  and  CEH  form  a  lune 
=  angle  C  x  2T. 

By  addition  we  get 
2(A+B  +  C)T  =  2  area  of  ABC  +  4T, 
and  (A  +  B  -f  C)  T  =  area  ABC  +  2T, 

and        (A  +  B  +  C)  T  —  2T  =  area  ABC. 

Area  ABC  =  (A  +  B  +  C  -  2)  T. 


BOOK     VIII. 


83 


THEOREM    XVI. 

The  area  of  a  spherical  polygon  is  equal  to  its  spher- 
ical excess  multiplied  by  the  trirectangiilar  triangle. 

Joining  A  and  C  and  A  and  D  by  arcs  of 
great  circles,  we  divide  the  polygon  into  tri- 
angles. 

The  sum  of  all  the  angles  of  the  triangles 
is  equal  to  the  sum  of  all  the  angles  of  the 
polygon  ;  hence,  the  area  of  the  polygon  is 
equal  to  the  sum  of  the  areas  of  the 
triangles. 

Let  S  =  Sum  of  all  the  angles, 

and  n  =  Number  of  the  sides  of  the  polygon, 

(n  —  2)  =         "        "  triangles. 
Area  of  polygon  =  [S  —  (n  —  2)  2]  T; 
Area  of  ABCDE  =  (S  -  %n  +  4)  T. , 


APPLICATION  OF  LOGARITHMS. 


A  Logarithm  is  the  exponent  or  power  of  some  number, 
which  is  called  the  base  of  the  logarithms. 

The  base  of  the  logarithms  in  common  use  is  10 ;  hence, 
10°  =  -fg-  =  1 ;  therefore,  the  logarithm  of  1        is  0. 

101  =  10;  "         «  *  "  10      "  1. 

102  =  100 ;  "  "  "  "  100     "  2. 

103  =  1000 ;  "  "  "  "  1000  "  3. 
The  integral  number  of  the  logarithm  is  called  the  Char- 
acteristic, and  when  positive   is   one   less  than  the  number 
of  integral  figures  in  the  number  of  which  it  is  the  logarithm. 

The  logarithm  of  any  number  between  1  and  10  must  be 
between  0  and  1 ;  the  logarithm  of  any  number  between  10  and 
100  must  be  between  1  and  2 ;  that  is,  the  logarithm  of  any 
number  hptwpp.n  1  and  10  is  a  fraction,  and  the  logarithm  of  any 
number  between  10  and  100  is  1  -f-  a  fraction. 

The  calculations  of  the  fractions  is  made  by  an  algebraic 
process,  and  logarithmic  tables  are  formed  to  facilitate  trigono- 
metrical computations.     Observe  also  that 

.1      =  —   =  10_1 ;  that  is,  the  logarithm  of  .1      is  —  1 ; 
and     .01    =^  =  10"2;      "    "    "  «         ".01    "-2. 

.001  =  ~  =  10~3;      "    "    «  "         "  .001  "  —  3. 

Rem. — The  characteristic  of  the  logarithm  of  a  fraction  is 
negative  and  corresponds  to  the  number  of  decimals ;  hence,  the 
logarithm  of  any  number  between  1  and  .1  is  between  0  and  —  1, 
and  is  put  —  1  -f-  a  fraction. 

As  logarithms  are  exponents,  they  can  only  be  used  as  such ; 
that  is,  for  Multiplication  and  Division,  for  Involution  and 
Evolution;  thus,  a1  x  a1  =  a2;  a2  x  «3  =  a5;  a1  x  b2  =  ab2\ 
a2  x  b2  =  a%& ;  a1  x  a1  x  a1  =  «8 ;  as  -^  a2  =  a1 ;  V«2  X  Vfl  — 
Va?  =  fl*5  VaM  =  ab^'y  V«W  =  <tfM;  a2bzc*cP-±aV<*d*e2  = 
ab-^cd-h~\ 


is 

3.656290 

a 

2.656290 

u 

1.656290 

a 

0.656290 

a 

1.656290 

u 

2.656290 

LOGAKITHMS.  85 

Rem. — Multiplication  is  performed  by  adding  the  exponents 
of  the  multiplier  and  multiplicand ;  Division  by  subtracting  the' 
exponents  of  the  divisor  from  those  of  the  dividend;  Involution 
by  multiplying  the  exponent  by  the  exponent  of  the  power  to 
which  it  is  to  be  raised ;  and  Evolution  by  dividing  the  expo- 
nents by  the  index  of  the  root. 

From  the  table  of  logarithms  the  decimal  parts  are  found, 
and  the  characteristic  must  be  added.  The  characteristic  of  an 
integral  number  will  always  be  one  less  than  the  number  of 
integral  figures. 

The  logarithm  of        4532. 

453.2 

"  "  45.32 

*  "  "  4.532 

"   .       "  .4532 

"  "  ,      .04532 

To  find  the  logarithm  of  a  number  of  three  or  less  figures, 

find  the  given  number  in  the  first  column  of  the  table,  and  take 

the  logarithm  under  zero ;  if  there  be  four  figures,  reserve  the 

one  in  the  units'  place  and  find  the  other  three  in  the  first  column, 

and  take  the  logarithm  under  the  figure  corresponding  to  the 

reserved  figure. 

The   logarithm  of  4530  is  3.656098 

"  "  45300  "  4.656098 

453000  "  5.656098 

To  find  the  logarithm  of  a  number  which  has  more  than  four 
figures: 

First  find  the  logarithm  of  the  number  expressed  by  the  four 
left-hand  figures ;  then  multiply  the  common  difference  given  in 
the  table  by  all  the  figures  left,  and  of  this  product  reject  as  many 
figures  on  the  right  as  there  were  reserved  figures,  and  add  the 
balance  to  the  logarithm  already  found. 

Thus,  find  the  logarithm  of  564236. 

The  characteristic  is  5. 

The  decimal  part  of  the  logarithm  of  5642  is  .751433 

Common  difference,  77  x  36  =  27 )  72  28 

.751461 
As  the  figures  cut  off  are  more  than  .5,  we  add  28  instead  of  27. 
To  which  add  the  characteristic  and  the 

logarithm  of  564236  =z  5.751461. 


86  LOGARITHMS. 

To  find  the  number  corresponding  to  the  given  logarithm  : 

If  the  given  logarithm  be  in  the  table,  take  the  three  figures 
opposite  in  the  first  column  and  the  one  immediately  over  it  at 
the  top  of  the  page,  and  point  off  one  more  integral  figure  than 
there  are  in  the  characteristic  of  the  logarithm. 

But  if  the  given  logarithm  is  not  in  the  table,  take  the  num- 
ber corresponding  to  the  next  less  logarithm  ;  then  divide  the 
difference  between  this  logarithm  and  the  given  one  by  the 
tabular  difference,  annexing  ciphers  to  the  dividend,  and  then 
affixing  this  quotient  to  the  number  already  found.  The  figure 
occupying  the  tenths'  place  must  take  the  first  place  after  the 
number  already  taken  from  the  table ;  this  will  sometimes  be  0. 


EXAMPLES. 

1.  Find  the  product  of  4573  and  6321. 

logarithm  4573  = 

3.660201 

logarithm  6321  = 

3.800786 

Next  less  number  (2890), 

7.460987 

Next  less  logarithm, 

898 

Difference, 

89 

Com.  diff.,  150.  • 

150  )  890  (  5933 

Hence,  number  corresponding,  28905933. 

2,  Divide  1728  by  12. 

logarithm  1728  == 

3.237544 

logarithm      12  = 

1.079181 

2.158363 
Number  corresponding,       144. 
There  will  be  slight  inaccuracies,  as  the  logarithms  are  only 
carried  to  six  places  of  decimals. 

3.  Find  the  logarithmic  sine  of  37°  15'  25". 

rin  37°  15'  =  9.781966 

sin  25"  (diff.  2.77  x  25  =  69.25)  =     69 

sin  37°  15'  25"  =  9.782035 

4.  Find  the  tangent  of  57°  30'  30". 

tang  57°  30'  ==  10.195813 

tang  30"  (4.65  x  30  =  139.50)  =  139 

tang  57°  30'  30"  =  10.195952 

I  will  not  add  any  more  examples,  as  the  learner  will  soon 
become  familiar  with  the  use  of  the  tables. 


TRIGONOMETRY. 


Trigonometry  treats  of  the  methods  of  finding  the  un- 
known parts  of  a  triangle,  when  certain  parts  are  known. 

Every  triangle  has  six  parts,  viz.,  three  sides  and  three  angles. 
When  three  of  these  are  known,  one  at  least  being  a  side,  the 
other  three  can  be  found  by  these  methods. 

Plane  Trigonometry  treats  of  plane  triangles. 

When  one  thing  depends  upon  another  for  its  value,  the  first 
is  said  to  be  a  function  of  the  secon'd. 

Sines,  tangents,  cosines,  cotangents,  etc.,  are  functions  of  arcs 
or  angles. 

DEFINITIONS. 

1.  The  Sine  of  an  angle  or  arc  is  a  line  drawn  from  the  end 
of  the  arc  perpendicular  to  the  radius  drawn  from  the  beginning 
of  the  arc  to  its  center,  or  to  the  vertex  of  the  angle. 

2.  The  Versed  Sine  is  the  distance  from  the  foot  of  the 
sine  to  the  beginning  of  the  arc. 

3.  The  Cosine  is  the  sine  of  the  complement  of  the  angle  ; 
it  is  equal  to  the  radius  minus  the  versed  sine. 

4.  The  Tangent  is  a  line  drawn  from  the  beginning  of 
the  arc  perpendicular  to  the  radius  at  its  extremity,  and  is 
limited  by  a  line  drawn  from  the  vertex  of  the  angle  through  the 
point  at  the  end  of  the  arc. 

5.  The  line  joining  the  vertex  of  the  angle  and  the  extremity 
of  the  tangent  is  called  the  Secant. 

6.  The  Cotangent  is-  the  tangent  of  the  complement  of 
the  angle. 


Vi 


88  PLAKE     TRIGONOMETRY. 

The  functions  of  aii  angle  are  exhibited  in  the  following 
diagram : 

BE  is  the  sine  of  the  acute. angle  ACB, 
/and  the  cosine  of  BCD,  its  complement.  i: * 

BF  or  CE  is  the  cosine  of  the  acute 
angle  ACB,  and  the  sine  of  BCD?  its  com- 
plement. 

^  AG  is  the  tangent  of  the  acute  angle 

ACB,  and  the  cotangent  of  BCD,  its  complement.    . 

DH  is  the  cotangent  of  the  acute  angle  ACB,  and  the  tangent 
of  BCD,  its  complement. 

The  radius  DC  is  the  sine  of  the  right  angle  ACD;  its  cosine 
is  zero;  the  tangent  would  be  AG  extended  until  it  meet  QD 
extended;  but  they  are  parallel;  hence  the  tangent  of  90°  is 
infinite,  and  the  cotangent  zero.  * 

If  the  angle  is  obtuse,  as  ACB',  its  sine  is  B'E',  the  same  as  of 
B'CK,  its  supplement.  All  the  other  functions,  except  the  versed 
sine  of  the  obtuse  angle,  will  be  the  same  length  as  of  the  acute 
angle,  its  supplement. 

All  the  functions  of  acute  or  right  angles  or  arcs  terminating 
in  the  first  quadrant  have  plus  signs.  If  the  angle  is  obtuse,  its 
cosine,  tangent,  and  cotangent  will  have  a  minus  sign. 

Cor.  l.« — The  sine  of  an  angle  cannot  be  greater  than  the 
radius  ;  but  the  tangent  may  have  any  value  whatever. 

Cor.  2. — It  is  needless  to  say  that  the  cosine  becomes  equal 
to  the  radius,  or  the  cotangent  infinite,  as  this  only  happens 
when  there  is  no  angle. 


PLANE     TRIGONOMETRY, 


89 


PROBLEM    I. 

To  show  the  relations  of  the  sines,  cosines,  tangents, 
and  cotangents  of  the  angles  of  a  right-angled  triangle 
with  the  side's  of  the  triangle, 

Let  PHB  be  a  triangle,  right- 
angled  at  H  ;  PF  or  PC  is  the  radius 
of  the  arc  CF ;  DC  is  the  sine  of  the 
angle  BPH,  PD  its  cosine,  and  FE 
its  tangent  and  cotangent  of  HBP, 
its  complement. 

The  triangles  PDC  and  PHB  are 
similar  ;  hence  the  proportions, 


CD:  BH  ::  PC:  PB ; 
that  is, 

sine  ?  \p  ::  R  :  h  ;        (1) 

sine  Px/t     ,'v 
•*.    P  = d—  >  (!) 


and 


R 

p  x  R 
sin~P  ' 


and      sine  P  =  —  r— 
h 

=  cosine  B, 
and  from  2  and  2, 

p  x  R  _  l_x_  R 
sin  P 


(2) 
(3) 

(3) 


and    PD:  PH  ::  PC:  PB; 
that  is, 

sine  B  :  b  : :  R  :  h  ; 

sine  Bx/t 


b  - 


and 


t.        b  x  R 

a  =  — - — 5" 
sm  B 


(1) 
(2) 


sine  B  = 


6xR 


=  cosine  P,      (3) 


and 


jB 


_  b_  ; 
sin  B ; 


(4) 
(4) 


sin  B  '  sin  P 

.-.    sin  P  :  sin  B   ::  p  :  b. 
Alternating  1st  and  2d  proportions, 

sin  P  :  R   ::  p  :  h,     and     sin  B  :  R   ::   b  :  h. 
The  triangles  PFE  and  PH B  are  similar ;  hence  the  proportion, 
EF  :  BH   ::    PF  :  PH  ;  that  is,  tang  P  :  p   ::    R  :  b; 

hence,  p  =  — L ,     and    b  =  -^^R — ->  by  analogy.  (5) 

Tang  P  =  ~^  =  cot  B,  and  tang  B  =  -—  =  cot  P,  an.,  (6) 


and 


Rxb 
tang  B' 


and    b  == 


Rxp 
tang  P 


(5) 


90  PLANE     TRIGONOMETKY. 

Eem. — In  a  right-angled  triangle,  either  acute  angle  is  a  com- 
plement of  the  other ;  hence,  the  sine  of  the  one  is  the  cosine  of 
the  other,  and  the  tangent  of  the  one  is  the  cotangent  of  the  other. 

As  these  formulas  are  general  principles  of  right-angled 
triangles  ;  they  must  be  fixed  indelibly  on  the  mind  of  the  student. 

Rem. — The  small  letters  represent  the  sides  opposite  the 
angles  having  corresponding  large  letters. 

FORMTJlAS. 

1.  Either  side  is  equal  to  the  sine  of  the  opposite  angle  mul- 
tiplied by  the  hypothenuse  and  divided  by  the  radius. 

2.  The  hypothenuse  is  equal  to  the  radius  multiplied  by  either 
side,  and  divided  by  the  sine  of  the  angle  opposite  this  side. 

3.  The  sine  of  either  acute  angle  is  equal  to  the  opposite  side 
multiplied  by  the  radius  and  divided  by  the  hypothenuse. 

4.  Either  side,  including  the  hypothenuse,  is  to  any  other 
side  as  the  sine  of  the  angle  opposite  the  former  side  is  to  the 
sine  of  the  angle  opposite  the  latter  side. 

5.  Either  side  is  equal  to  the  tangent  of  its  opposite  angle 
multiplied  by  the  other  side  and  divided  by  the  radius  ;  or, 

Either  side  is  equal  to  the  radius  multiplied  by  the  other  side 
and  divided  by  the  tangent  of  the  angle  opposite  this  other  side. 

6.  The  tangent  of  either  acute  angle  is  equal  to  its  opposite 
side,  multiplied  by  the  radius  and  divided  by  the  other  side. 

EXAMPLES. 

1.  Given  k  =  43,  B  =  25°,  to  find;?,  b,  and  P. 

p  =  90°  — 25°  =  65°. 

sin  P  ^  If 

Formula  1,  p  = ~ ;  .*.  logjp=logsin65°  +  log43— log  R, 

K 

log  siu  of  65°  =s     9.957276 

log  43  — 10  =     1.63346.8 

p  =  38.971,  No.  corres.  1.590744 

Formula  1,  b  -as        „ —  ;  .*.  log.  b  =  log  sin  25°+ log  43— log  R, 

log  sin  25°  =     9.625948 

log  43-10  =     1.633468 

b  =  18.172,  No.  corres.  to  log,      1.259416 

2.  Given  h  =  624,  P  =  48°,  to  find;?,  b,  and  B. 

B  =  42°,  p  ±=  463.723,  and  b  =  417.538,  Ana. 


PLANE     TRIGONOMETfiY. 


91 


3.  Given  b  =  535,  B  =  65°  15',  to  find  P,p,  and  k. 

P  =  24°  45',  p  ss  246.738,  and  h  =  589.114,  Ans. 

4.  Given  b  =  47,  P  =  35°,  to  find  -B,  p,  and  A. 

B  ==  55°, p  =  32.91,  and  h  =  57.376,  <4ns. 
5;  Given./?  =  275,  B  =  58°,  to  find  P,  b,  and  li. 

P  b=  32°,  b  s=  440,92,  and  A  =  518.946,  ^«* 
6.  Given  p  =  15,   J  =  25,  to  find  P,  B,  and  h.     (Use  for- 
mula 6.)     7*  =  29.05,  B  =  59°  23'  3 ",  and  P  =  30°  36'  57",  Ans. 

PROBLEM    II. 

To  show  that  the  sines  of  the  angles,  in  any  plane 
triangle,  are  respeetively  proportional  to  their  opposite 
sides. 

Let  ABC  be  any  plane  triangle,  and     • 
from  any  vertex  as  B  draw  a  perpendic- 
ular BD  to  the  opposite  side  AC,  making 
two   right-angled   triangles,  ABD    and 
BCD;  from  which  we  have  in  the  tri- 


angle ABD,  BD 


triangle  BCD,  BD  = 


sin  A  x  c       ,  .     ,, 
g, — ,  and  in  the 

smCxff 


R 


.  sin  A  x  c  =  sin  C  x  a ;  hence, 


sin  A  :  sin  C  ::  a  :  e,  and  by  inversion,  sin  C  :  sin  A  ::  c  \  a, 
and  sin  B  :  sin  A  :  :  b  :  a,  and  sin  A  :  sin  B  :  :  a  :  b; 
hence,  in  any  triangle,  the  sines  of  the  angles  are  respectively 
proportional  to  their  opposite  sides. 


1.  Given  A 


EXAM  PLES. 

55°,  B  =  51°,  c  =  143,  to  find  C,  a,  and  b. 
C  ss  74°,  a  =  121.86,  and  b  =  151.61,  Ans. 
2.  Given  A  =  60,  c  =  54,  and  A  ==  26°,  to  find  B,  C,  and  b. 
C  =  23°  10'  13",  B  =  130°  45'  47",  and  b  =  103.667,  Ans. 

Rem. — If  c  is  greater  than  a 
there  is  two  triangles;  thus,  c.  = 
60,  a  =  54,  and  A  ==  26°. 

Observe,  that  the  angle  AC'B  is 
the  supplement  of  ACB. 

C  =  29°  8'  56",  C  =  150°  51' 
4",  B  =  124°  51'  4",  B'  *=  3°  .8' 
56",  b  =  101.089,  and  b'  =  6.7665,  Ans. 


92 


PLANE     TRIGONOMETRY 


/     PROBLEM    III. 

To  show  by  Diagram  No.  2  certain  relations  of  the 
functions  of  angles,  when  the  radius  is  unity. 

In  the  triangle  CEB, 

BE2  +  CE2  =  CB2; 
that  is,        sin2  C  +  cos2  C  =  R2  =  1, 

sin2  C  =  1  —  cos2  C, 
and  cos2  C  =  1  —  sin2  C. 


0) 

(2) 
(3) 

The  triangles  CEB  and  CAT  are  similar,  and  also  CFB  and 
CDT'  are  similar. 


.-.     CE  :  CA  ::  EB  :  AT, 
cos  C  :  1  : :  sin  C  :  tang  C, 
cos  C  x  tang  C  =  sin  C,        (4) 
,        ^       sin  C        /rN 

tanec  =  ^s-c;    (5> 

tang  C  x  cot  C 


.%    CF  :  CD  ::  FB  :  DT', 
sin  C  :  1  : :  cos  C  :  cot  C, 
sin  C  x  cot  C  =  cos  C, 
cos  C 
sin  C* 


cot  C 


(6) 

(?) 


sin  C        cos  C       .. 
cos  C        sin  C 


tan  C  = 
cot  C 


(8) 
(9) 


cot  C 

1 
tang  C 

Rem. — In  the  above  diagram,  C  represents  any  angle ;  hence 
the  relations  apply  to  all  angles. 


SYNOPSIS    OF    ^HE    FORMULAS, 

cin2 


I 


cosH=  1. 

(i) 

sin2  =1  —  cos2. 

'  m 

cos2  =  1  —  sin2. 

(3) 

sin  =  cos  x  tang. 

(4) 

sin 

tang  = 

°    cos 

(5) 

cos  ==  sin  x  cot. 

(6) 

cos 

cot  =  - — 

sm 

(?) 

*"•*  =  m 

(8) 

COt  = • 

tang 

(9) 

PLANE     TRIGONOMETRY 


93 


PROBLEM    IV. 

To  show  the  relations  of  the  functions  of  a  right- 
angled  triangle  with  the  sides,  when  the  radius  is 
unity. 

The  triangles  PDC  and  PHB  are 
similar,  and  give  the  proportions, 


CD  :  BH 

and        PD  :  PH 

sin  P  :  p 

sin  B  :  b 


PC  :  PB, 
PC  :  PB. 
1  :  A, 
1  :  lu 


p  =  sin  P  x  h, 

sin  P 

sin  P  =  ?, 
h 


V 


b  —  sin  B  x  h,  (1) 

_b_ 
sin  B' 

b 


h  = 


sin  B  _ 
h 


(3) 

(3) 


From  (2),    —  -£-=  =  - — =,        .-.    sin  P  :  sin  B   ::  p  :  b. 
v  '     sin  P       sin  B 

The  triangles  PFE  and  PHB  are  similar; 
BH  :  EF    ::    PH  :  PF, 
p  :  tang  P    : :    b  :  1 ; 


hence, 

and  by  a  similar  process, 


p  =  tang  P  x  b, 
b  =  tangP' 


and 


b  =  tang  B  x  p, 

b 
P  ~  tangB' 

tang  P  =  |  =  cot  B, 


tang  B  =  -  =  cot  P. 
P 


(4) 


(5) 


94 


PLANE     TRIGONOMETRY. 


ENUNCIATION    OF    FORMULAS. 

1.  Either  side  is  equal  to  the  product  of  the  sine  of  the  oppo- 
site angle  and  the  hypothenuse. 

2.  The  hypothenuse  is  equal  to  either  side,  divided  by  the 
sine  of  the  angle  opposite  that  side. 

3.  The  sine  of  either  acute  angle  is  equal  to  its  opposite  side 
divided  by  the  hypothenuse. 

4.  Either  side  is  equal  to  the  product  of  the  tangent  of  its 
opposite  angle  and  the  other  side ;  or,  either  side  is  equal  to  the 
other  side  divided  by  the  tangent  of  the  angle  opposite  that 
other  side. 

5.  The  tangent  of  either  angle  is  equal  to  its  opposite  side 
divided  by  the  other  side. 

Rem. — In  computing  these  formulas  by  logarithms,  when  the  formula 
consists  of  the  product  of  two  numbers,  10  must  be  subtracted  from  the  sum 
of  the  logarithms  ;  and  when  it  is  fractional,  10  must  be  added  to  the  differ- 
ence of  the  logarithms. 


PROBLEM    V. 

To  find  the  sine  and  cosine  of  the  sum  and  difference 
of  two  arcs,  whose  sines  and  cosines  are  known. 


Let  ACB  and  BCD  be  the  two  angles 
whose  sines  and  cosines  are  known. 

Let  the  angle  ACB  be  designated  angle  A. 

B. 


« 

"     BCD 

i(              a 

Make 

BCE  = 

BCD; 

then 

DK  = 

sin  (A  +  B), 

and 

EM  = 

sin  (A  — 

B). 

CK  = 

--  cos 

(A  +  B), 

CM  = 

:  COS 

(A-B). 

and 

The  triangles  DFG  and  GHE  are  equal  and  similar  to  CLG, 
whose  sides  are  respectively  perpendicular  to  those  of  DFG 
and  GHE. 

the  angle  FDG  =  angle  LCG  =  angle  A. 


PLANE     TRIGONOMETRY.  95 

In  the  triangle  DFG, 

DF  =  DG  x  cos  A, 
and  in  the  triangle  CLG, 

GL  =  CG  x  sin  A. 
DF  =  sin  B  x  cos  A, 
and  GL  =  cos  B  x  sin  A, 

GL  +  DF=  DK  =  sin  (A+B)  =  sin  Ax  cos  B  4-  cos  A  xsin  B  )  & 
G  L  —  DF  =  EM  =  sin  (A—  B)  =  sin  A  x  cos  B  —  cos  A  x  sin  B  \  $ 
By  addition, 

sin  (A  +  B)  4-  sin  (A  —  B)  =  2  sin  A  x  cos  B, 
and         sin  (A+B)  —  sin  (A  —  B)  ±=  2  cos  A  x  sin  B. 

Put  A  +  B  =  M,        and        A  -  B  =  N ; 

then  A  =  £(M  +  N),      and      B  =  £(M-N); 


then      sin  M  -f- sin  N  =  2  sin|(M  +  N)  x  cos -J-(M  —  N), 

(1) 

and        sin  M  —  sin  N  =2  cos£(M  +  N)  x  sin£(M  — N); 

(2) 

dividing  (1)  by  (2),  and  reducing  by        tang  =  — , 

cos 

sinM  +  sinN       sin£(M  +  N)  xcos£(M  —  N)'      tang|(M  +  N) 
sin  M  —  sin  N  —  cos  J(M  -i-N)  xsin  £(M  — N)  —  tangi(M  — N) 

(3) 

In  the  triangle  CLG,     CL  =s  CG  x  cos  A  =  cos  B  x  cos  A. 
"  "        DFG,     FG  =  DG  x  sin  A  =  sin  B  x  sin  A. 

By  subtraction  and  addition, 

CL  —  FG  =  CK  =  cos  M  =  cos  A  x  cos  B  —  sin  A  x  sin  B  )  S> 
CL  +  FG  =  CM  —  cos  N  =  cos  A  x  cos  B  -f-  sin  A  x  sin  B  \  1 

Cor. — By  addition  and  subtraction, 
cos  M  +  cos  N  =  2  cos  -J-  (M  +  N)  x  cos  i  (M  —  N),       (4) 
cos  M  —  cos  N  =  —  2  sin  |(M  +  N)  x  sin  £(M  —  N) ;  (5) 
dividing  (5)  by  (4), 

cos  M  —  cos  N        —  sin  \  (M  -f  N)       sin  J(M  —  N) 
cos  M  +  cos  N  ~      cos  \  (M  +  N)         cos  J  (M  —  N) 

=  -  tang  i  (M  +  N)  x  tang  \  (M  -  N). 

Observe  that  the  tangent  has  a  negative  sine,  which  is  correct, 
as  the  numerator  of  the  first  member  of  the  equation  is  negative, 
the  cosine  N  being  greater  than  the  cos  M ;  a  small  angle  has  a 
larger  cosine  than  a  large  angle. 


PLAXE     TRIGONOMETRY. 


EXAM  PLES. 


1.  To  find  the  sine  and  cosine  of  30°,  60°,  and  45°,  when  the 
radius  is  1. 

The  sine  of  30°  is  half  the  chord  of  60°,  the  chord  of  60°  is 
radius  =  1 ;  hence,  sine  of  30°  =  £,  and  cosine  of  60°  =  £.        t 

sin  30°  =  cos  60°  =  J. 

Cosine  of  30°  =  VT^J  =  V%  =  W$'>  sme  of  60°  =  .f\/3; 
and  cos  30°  =  sin  G0°  =  |\/3. 

When  the  angle  is  45°,  the  sine  and  cosine  will  be  equal,  and 
as  sin2  -f  cos2  =  1,  that  is,  f-'-K-J  ==  1. 

sin2  45°  =  i      and      sin  45°  =  Vi  =  W%  =  cos  45°, 

and  sin  45°  =  cos  45°  =  Ja/2. 

2.  To  find  the  sine,  cosine,  tangent,  and  cotangent  of  every 
arc  from  1°  to  90°. 

The  semi -circumference,  when  the  radius  is  1,  is  3.1415926535 ; 
which  being  divided  by  10800,  the  number  of  minutes  it  contains, 
gives  the  length  of  1',  equal  to  .0002908882,  which  in  so  small  an 
arc  does  not  differ  materially  from  the  sine  of  1',  and  may  be 
regarded  as  such;  and  cos  1'  =  Vl~ sin*!'  =  .9999999577. 

By  taking  the  formula  from  Prob.  5, 

sin  (A  -f  B)  -+-  sin  (A  —  B)  =  2  sin  A  x  cos  B, 

by  transposition, 

sin  (A  -h  B)  =  2  sin  A  x  cos  B  —  sin  (A  —  B), 

and  making  B  =  1',  and  A  =  1',  2',  3',  4',  etc.,  in  succession, 

sin  1'  =  .0002908882; 

sin  2'  =  2  sin  V  x  cos  1'  -  sin  0   =  .0005817764; 

sin  3'  =  2  sin  2'  x  cos  1'  —  sin  1'  =  .0008726646. 


PLANE     TRIGONOMETRY.  97 

By  continuing  this  process,  we  can  get  the  sines  of  every  arc 
from  1'  to  90° ;  and  taking  them  in  an  inverse  order,  we  have  the 

cosines  of  every  arc  from  1'  to  90°  ;  then,  as  tang  =  — ,  we  can 

get  the  tangents  of  every  arc  from  1'  to  90° ;  and  by  taking  them 
in  an  inverse  order  we  have  the  cotangents  of  every  arc  from  1' 
to  90°.  These  will  form  a  table  of  natural  sines,  cosines,  tangents 
and  cotangents. 

The  logarithms  of  these  numbers,  with  the  addition  of  10  to 
to  each  logarithm,  forms  the  table  of  logarithmic  sines,  cosines, 
etc.  In  the  table  the  radius  is  taken  as  ten  billions,  whose  log- 
arithm is  10 ;  and  as  the  functions  are  proportional  to  the  radii, 
hence  the  natural  sines,  etc.,  must  be  multiplied  by  this  number, 
which  is  done  by  adding  the  logarithms  of  the  natural  sine,  etc., 
and  of  the  radius. 


PROBLEM    VI. 

Two  sides   and  the   included   angle   of  a   triangle 
given,  to  find  the  other  angles. 

By  Problem  2, 

a  :  b    : :    sin  A  :  sin  B ; 

by  composition  and  division, 

a  +  b  :  a  —  b    : :    sin  A  +  sin  B  :  sin  A  —  sin  B, 

a  +  b sin  A  +  sin  B 

a  —  b  ~~  sin  A  —  sin  B 

By  Problem  5,  Formula  (3), 

sin  M  -f  sin  N  _  tang  £  (M  -f  N)  # 
sin  M  —  sin  N  ~  tang  i  (M  —  N) ' 

a  +  b  _  sin  A  +  sin  B  _  tang  \  (A  -f  B)  # 
a  —  b  ~~  sin  A  —  sin  B  ~~  tang  £  (A  —  B) ' 

hence  the  proportion, 

a  +  b  :  a  —  b    ::    tang  J(A  +  B)  :  tang  £  (A  —  B)- 

Knowing  the  sum  and  difference  of  two  angles,  we  easily  find 
the  angles. 

5 


98 


PLANE     TRIGONOMETRY. 


PROBLEM  VII. 

To  find  the  area  of  a  triangle,  having  given   two 
sides  and  the  included  angle. 

The  angle  A  and  the  sides  b  and  c 
given. 

area  ABC  =  \pc. 

In  the  triangle  ADC, 

p  =  sin  A  x  o ; 

area  of  ABC  =  |5  x  c  x  sin  A. 


PROBLEM    VIII 

7/*  from  the  vertex  of  any  angle  of^a  triangle  a  line 
be  drawn  perpendicular  to  the  opposite  side,  produced 
if  necessary,  then  will  the  sum  of  the  segments  of  the 
opposite  side  he  to  the  sum  of  the  other  two  sides  as  the 
difference  of  those  sides  is  to  the  difference  of  the  seg- 
ments. 

p2  =  a2  —  n2  =  c2  —  m2, 

m2  —  n2  =r  c2  —  a2, 

(m  +  n)  {m  —  n)  =  (c  +  a)  (c  —  a), 

m  -f-  n  :  c  -\-  a    ::    c  —  a  :  m—  n. 


PROBLEM    IX. 

If  from  the  half  sum  of  the  three  sides  of  a  triangle, 
each  side  be  subtracted  separately,  then  the  square  root  of 
tlie  continued  product  of  the  half  sum  and  the  tliree 
remainders  will  be  the  area  of  the  triangle. 

*-  c2- 
c2 


f 


nv 


a2  —  n2, 


and 


a<  =z  m 


2  —  V)2 


m2  —  n2  =  c2  —  a2, 
(m  +  n)  (m  —  ri)  =  c2  —  a2 ; 


PLANE     TRIGONOMETRY.  99 


and  as  m  +  n  =  b, 

b  (m  —  n)  =  c2  —  a2, 

&  —  a2 
m  —  n  =  — r — , 

m  +  n  =  b, 


2m  =  b  +  & 


m  = 


b      ' 
b2  +  &  +  a2 


A  2  2  2  .  (P  +  P  —  tfY 

and  p2  =  c2  —  m2  =  c2  —  I -^ ) , 


46V-  (y  +  c2  -a2)2 

4J2 


+  $2  +  g2  _  ^2)  [2&c  —  (b2  +  c2  —  a2)] 
ib2" 

[(F+"^-fl2]  x  [fl2-"^^)2] 
4£2 

/(b+c  +  a)  (b  +  c—a)(a  +  b—c)(a  +  c—b) 
=  Y  4^ 

area  of  ABC  =  \pb 

/(b  +  c  +  ajJb-\-c^~a)(a  +  b—c)  {a  +  c—b)  [& 

=  y w  x  V  4  ; 

b2  is  canceled,  and  the  two  4's  factored, 
area  of  ABC  -  -  /(P+c+dHb+c-<^±b-41a+c^L 


=  ,  /&  +  <>  +  «) 


2  2  2 


EXAMPLES. 

Two  sides  and  the  included  angle  given. 
1.  Given  a  =  75,  5  =  90,  and  C  ==  20°,  to  find  A,  B,  and  e. 
b  +  a  :  b  —  a  : :   tang.  J  (B  -f  A)  :  tang.  £  (B  —  A) ; 
165  :  15  : :  tang.  80°  :  tang.  B  —  A. 

tang,  t  (B  —  A)  =  log  15  +  log  tang.  80°  —  log  165, 

■L  (B  +  A)  =    80°  00'  00", 
j(B  — A)  =    27    16  27. 

B  =  107°  16'  27", 

A  =    52°  43'  33". 


100  PLANE     TRIGONOMETRY. 

Sin  A  :  sin  C  : :  a  :  c, 
log  c  =  log  sin  C  +  log  a  —  log  sin  A. 

Arith.  comp.  165  =    7.782516 

log  15  =    1.176091 

log  tang.  80°  =  10.753681 

tang.  \  (B-A)  =  27°  16'  27"  =    9.712288 

Arith.  comp.  sin  52°  43'  33"    =    0.099225 
log  sin  C  20°  9.534052 

log  a  75  1.875061 

c  =  32.235  1.508338 


When  the  three  sides  are  given. 

2.  Given    a  =  237,      I  =  495,     and 
C  =  327. 

m  -f  n  :  c  +  a  ::  c  —  a  :  m—n; 
m  4-  n  =  h, 
log  {m  —  n)  =  log  (c  +  a)  +log  (c  —  a) 
—  log  I. 

m  +  n  =  495 
m  —  n  =  102.546 
m  =298^773, 

and  n    =  196.227. 


In  the  triangle  ABD, 


c  :  m   ::    R  :  sin  ABD  ; 
log  sin  ABD  =  log  R  +  log  m  —  log  c. 


Arith.  comp.  495 
log  564 
log  90 

(m  -  n)  =  102.546 

Arith.  comp.  327 

log  298.773 

logR 

sin  ABD  =  cos  A  23°  59' 


=  7.305395 
=  2.751279 
=    1.954243 


2.010917 

7.485452 

2.475342 

10.000000 

9.960794 


PLANE     TRIGONOMETRY.  101 


In  the  triangle  BDC,    a  :  n  ::   R  :  cos  C; 

log  cos  C  =  log  n  +  log  R  —  log  a. 


A  =  23°  59'  00" 
C  =  34°    6'  36" 


180  —  58°    5'  36"  =  121°  54'  24"  =  B. 

Arith.  comp.  237  =    7.625252 
log  196.227  =    2.292759 

log  R  10.000000 

cos  C  34°  6'  36"  9.918011 


PKOBLEM. 

A  side  and  two  adjacent  angles  given,  also  two  sides 
and  an  angle  opposite  one  of  them. 


EXAM  PLES. 

1.  Given  A  =  32°,  a  =  40,  and  b  =  50,  to  find  B,  C,  and  c. 

(  B  =    41°  28'  59",  C  =  106°  31'    1",  and  c  =  72,368. 
Ans.  |  B  _  138o  31,    j-,r  c  =      9o  2g,  g9„     „    c  _  12#436# 

In  this  case  there  are  two  triangles. 

2.  Given  a  =  450,  b  =  540,  and  C  ==  80°,  to  find  A,  B,  and  c. 

A  =  43°  49',  B  =  56°  11',  and  c  =  640.08. 

3.  Given  a  =  40,  b  =  34,  and  c  =  25  yards,  to  find  the 
angles.    A  =  83°  53'  16",  B  =  57°  41'  24",  and  C  =  38°  25'  20". 

4.  Given  b  =  306,  c  =  274,  and  B  =  78°  13',  to  find  A,  C, 
and  a.  A  =  40°  33',  C  s=  61°  14',  and  a  =  203.2. 

5.  Given  B  =  100°,  a  -  280.3,  and  c  =  304,  to  find  A,  C, 
and  b.  A  =  38°  3'  3",  C  =  41°  56'  57",  and  b  =  447.856. 

6.  Find  the  area  of  a  triangle  having  two  sides  equal  to  30 
and  40  ft.  respectively,  and  the  included  angle  28°  57', 

Ans.  290.427  sq.  ft. 

7.  Find  the  area  of  a  triangle  whose  sides  are  respectively  30, 
40,  and  50  rods.  Ans.  3  acres  3  rods. 

8.  What  is  the  area  of  a  triangle,  whose  base  is  50  rods  and 
altitude  30  rods  ?  Ans.  4  acres  2  rods  30  perches. 


102 


PLANE     TEIGONOMETEY, 


PRACTICAL    PROBLEMS. 


1.  Find  the  distance  AC  across  a  deep 
river,  having  given  AB  =  500  yards,  the 
angle  BAC  =  74°  14',  and  the  angle  ABC  = 
49°  23'. 

sin  C  :  sin  B   : :   c  :  h ; 
log  b  =  log  sin  B  +  log  c  —  log  sin  C 
=  577.8  yards. 

2.  Given  AC  =  735  yards,  BC  =  840, 
and  the  angle  C  =  55°  40',  to  find  AB. 

Two  sides  and  the  included  angle. 

AB  ==  741. 

3.  Given  AB  =  600  yards,  and  the  adja- 
cent angles  A  =  57°  35'  and  B  =  64°  51',  to 
find  the  angle  C  and  the  sides  AC  and  BC. 

AC  =  643.49  yards. 
BC  =  600.11      " 


4.  Find  the  height  of 
D  a  point  on  a  mountain 
above  a  horizontal  plane. 
The  angle  of  elevation  at 
B,  a  point  at  the  foot  of 
the  mountain,  is  27°  29' ; 

and  at  A  distant  from  B  975  yards,  in  a  direct  line  from  B,  and 
in  the  plane  DBA,  is  15°  36'. 

DC  =  587.61  yards. 

5.  Wishing  to  know  the  distance  between 
two  inaccessible  objects  C  and  D,  I  measured 
a  line,  AB  =  339  feet,  from  both  ends  of  which 
the  objects  were  visible ;  I  found  the  angles 
BAD  =  100°,  BAC  =  36°  30',  ABC  =  121°, 
and  ABD  =  49°;  find  the  distance  DC. 

DC  =  697J  feet. 


---^ 


PLAtfE     TRIGONOMETRY. 


ioa 


6.  Wishing  to  know  the  dis- 
tance between  two  inaccessible 
objects,  A  and  B,  and  finding  no 
place  from  which  both  could  be 
seen,  two  points  C  and  D,  200 
yards  distant,  were  found;  from 
the  former  point  A  could  be  seen, 

and  from  the  latter  B ;  from  C,  a  distance  of  200  yards  were 
measured  to  a  point  F,  from  which  A  could  be  seen  ;  and  from  D 
the  same  distance  was  measured  to  E,  from  which  B  could  be 
seen,  and  the  following  angles  taken,  viz., 


ACD  =  53°  20', 
ACF  =  54°  31', 
AFC  =  83°  00', 

Find  the  distance  AB. 


BDC  =  156°  25', 
BDE  =  54°  30', 
BED  =    88°  30'. 

Ans.   AB  —  345.467  yards. 


7.  The  distance  between  three  points 
A,  B,  and  C,  are  known,  viz., 


and 


AB 
AC 
BC 


800  yds., 
600    " 
400    " 


All  are  visible  from  a  distant  point  P 
at  which  the  angles  are  measured, 

A  PC  =  33°  45', 

Find  AP,   BP,  and  CP 


and 


BPC  =  22°  30'. 


Ans.    AP  =    710.193  yds., 
BP  =    934.291    " 
CP  =  1042.522    " 


SPHERICAL  TRIGONOMETRY. 


Spherical  Trigonometry  treats  of  spherical  triangles, 
the  sides  of  which  are  arcs  of  great  circles,  each  less  than  180°, 
and  the  angles  are  diedral  angles,  formed  by  the  planes  of  the 
great  circles  ;  each  angle  is  less  than  two  right  angles. 


Napier's  Five  Circular  Parts  form 
the  basis  for  the  analysis  of  the  functions  of 
right-angled  spherical  triangles. 

The  two  sides  about  the  right  angle,  and  the 
complements  of  the  hypothenuse  and  of  the  two 
oblique  angles  are  the  five  circular  parts. 

The  spherical  triangle  ABC  is  right-angled 
at  A.     The  sides  b  and  c,  and  the  complements 
of  the  hypothenuse  a  and  of  the  angles  B  and  C 
are  the  five  circular  part3. 

In  taking  any  three  of  these  parts,  they  will  either  be  found 
to  be  adjacent  to  each  other,  or  one  of  them  will  be  separated 
from  both  the  others.  When  they  are  adjacent,  the  one  lying 
between  the  others  is  called  the 
middle  part,  and  when  they  are 
not  adjacent,  the  one  separated 
from  both  the  others  is  the  middle 
part  and  the  others  are  opposite. 

Let  ABC  be  a  spherical  tri- 
angle, right-angled  at  A,  0  the 
center  of  the  sphere.  Draw  CD 
perpendicular  to  OA,  and  DE  per- 
pendicular to  OB,  and  join  CE.  As  the  angle  A  is  a  right  angle, 
the  angle  CDE  is  also  a  right  angle,  as  CD  is  perpendicular  to  the 


SPHERICAL     TRIGONOMETRY.  105 

plane  ABO  in  which  DE  is  drawn  perpendicular  to  OB,  a  line  of 
the  plane;  hence  CE  is  perpendicular  to  OB  (Th.  3,  Bk.  6),  and 
CED  =  B. 

1.  sin  b  =  sin  a  x  sin  B  =  cos  comp.  a  x  cos  comp.  B. 

2.  sin  c  =  sin  a  x  sin  C  =  cos  comp.  a  x  cos  comp.  C. 

3.  cos  B  =  cos  b  x  sin  C  =  cos  b  x  cos  comp.  C  =  sin  comp.  B. 

4.  cos  C  s=  cos  c  x  sin  B  =  cos  c  x  cos  comp.  B  =  sin  comp.  C. 

5.  cos  a  =  cos  b  x  cos  c  =  cos  b  x  cos  c  =  sin  comp.  a. 

By  Prob.  4,  in  the  triangle  CED, 

CD  =  CE  x  sin  B, 

sin  b  =  sin  a  x  sin  B.  (1) 

No.  2  is  derived  in  the  same  way,  by  making  B  the  vertex 

instead  of  C. 

d        DE         DE 

cos  B  =  ^  =  - 

CE        sin  a 

DE  =  cos  b  x  sin  a. 

sin  c  =  sin  a  x  sin  C.  (2) 

In  the  triangle  OED, 

DE  =  OD  x  sin  DOE, 

D       cos  b  x  sin  a  x  sin  C  D        .    -  /oX 

.-.     cos  B  = s =  cos  B  x  sin  C.         (3) 

sin  a  v  ' 

In  No.  4,  cos  C  is  found  like  cos  B,  by  making  B  the  vertex. 
In  the  triangle  ODE, 

OE  =  ODx  cos  DOE; 
that  is,  cos  a  —  cos  b  x  cos  c,  (5) 

From  these  five  formulas,  five  others  may  be  derived  ;  thus, 

^        .     ,  .  .     _      sin  c   x  cos  C      sin  c      cos  C 

1.  sin  b  =  sin  a  x  sin  B  =  —  — ^ == x  - — =; 

sin  C  x  cos  c       cos  c      sin  C 

==  tang  c  x  cot  C  =  sin  b. 

0  .  .     _       sin  b   x  cos  B       sin  b      cos  B 

2.  sin  c  =  sin  a  x  sin  C  =  - — =-- r  = 7  x  -. — 5 

sin  B  x  cos  J       cos  6       sin  B 

=  tang  b  x  cot  B  =  sin  c. 


106  SPHERICAL     TRIGONOMETRY. 

~  0  7        .     ~      cos  a  x  sin  c       cos  a       sin  c 

3.  cos  B  =  cos  b  x  sm  C  = — : =  - x 

cos  c  x  sm  a       sin  a       cos  c 

=  tang  c  x  cot  a  =  sin  comp.  B. 

.  ^  .     _>       cos  a  x  sin  b        sin  #        cos  a 

4.  cos  C  =s  cos  c  x  sin  B  = -, r —  = T  x  - — 

cos  b  x  sin  a       cos  0       sm  a 

=  tang  5  x  cot  a  =  sin  comp.  C. 

-  ,  cos  B  x  cos  C      cos  B      cos  C 

5.  cos  a  =  cos  o  x  cos  c  =  -* — ^ : — =  =  -s — =>  x  -s — ^ 

sin  C  x  sm  B       sm  B      sm  C 

=  cot  B  x  Cot  C  =  sin  comp.  a. 

From  the  first  five  formulas : 

The  sine  of  the  middle  part  is  equal  to  the  produot  of  the  sines 
of  the  opposite  parts. 

From  the  second : 

The  sine  of  the  middle  part  equals  the  product  of  the  tangents 
of  the  adjacent  parts. 

Rem. — Observe  that  the  cosine  of  an  angle  is  equal  to  the 
sine  of  the  complement,  and  the  cotangent  is  equal  to  the  tangent 
of  the  complement. 


THE    SPECIES    OF    THE    FUNCTIONS    OF    ANGLES 

OR    ARCS. 

As  the  functions  of  an  arc  and  of  its  supplement  are  lines  of 
equal  length,  there  is  a  distinction  necessary,  in  order  that  we 
may  know  whether  the  arc  is  greater  or  less  than  90°  ;  hence  the 
minus  sign  is  given  to  the  co"sine,  the  tangent,  and  the  cotangent 
when  the  arc  is  greater  than  90°,  or  terminates  in  the  second 
quadrant. 

Two  arcs  are  said  to  be  of  the  same  species  when  they  are 
both  less  or  both  greater  than  90°,  and  of  different  species  when 
the  one  is  greater  and  the  other  less  than  90° 

1.  From  the  3d  and  4th  formulas  of  circular  parts, 

.    n       cos  B  ,  .     _       cos  C 

sm  C  = r  i         and         sin  B  = 

cos  b  cos  c 

As  the  sines  of  C  and  B  are  both  positive,  hence  the  cosines 
of  each  oblique  angle  must  have  the  same  sign  as  the  cosines  of 


SPHERICAL     TRIGONOMETRY.  107 

the  opposite  sides ;  consequently,  the  oblique  angles  and  their 
opposite  sides  are  of  the  same  species. 


2.  When  the  hypothenuse  is  less  than  90°,  the  other  two  sides 
and  their  opposite  angles  are  of  the  same  species ;  for,  as 

cos  a  =  cos  b  x  cos  c, 

and  when  a  is  less  than  90°  its  cosine  is  positive;  hence  the 
cosines  of  b  and  c  have  like  signs,  that  is,  b  and  c  are  of  the  same 
species.  But  when  a  is  greater  than  90°,  its  cosine  is  negative ; 
hence  the  cosines  of  b  and  c  have  different  signs ;  that  is,  b  and 
c  are  of  different  species. 


By  these  two  rules  the  nature  of  each  result  is  determined, 
except  when  an  oblique  angle  and  the  opposite  side  are  given,  to 
find  the  other  parts. 

Let  ABC  be  right-angled  at  A ; 
and  B  and  b  be  known. 

1st.  If  the  sine  of  b  is  greater 
than  the  sine  B,  there  can  be  no 

solution ;  for,  as  sin  a  =  -: — ~>1, 
sin  B 

which  is  impossible. 

2d.  If  sine  b  =  sin  B,  then  sin  a  =    .     -»  =  1 ;  hence,  the 

sin  B 

vertex  B  is  the  pole  of  the  opposite  side  b,  and  a  and  c  are 

each  90°. 

3d.  If  sine  b  is  less  than  sine  B,  when  B  is  less  than  90°, 
there  will  be  two  solutions,  as  shown  in  the  above  figure;  as  ABC 
and  AB'C  both  fulfill  the  conditions.  When  B  is  greater  than 
90°  ;  then,  in  order  that  sin  b  <  sin  B,  the  side  b  must  be  greater 
than  the  angle  B ;  when  the  result  will  be  the  same  as  above, 
and  a  and  c  in  the  one  triangle  will  be  complements  of  the  same 
letters  in  the  other  triangle. 


108  SPHEEICAL     TRIGONOMETRY. 

EXAMPLES. 

1.  Given  a  =  86°  51',  and  B  =  18°  3'  32",  to  find  b,  c,  and  C. 

1.  sin  b  =  sin  a  x  sin  B, 
5.  cos  c  =  cos  a  -f-  cos  b ; 
4.  cos  C  =  cos  c  x  sin  B. 

log  sin  B  =  18°    3'  32"  =  9.491354 

cos  c         =  86°  41'  14"  =  8.761826 

C  _  10     =  88°  58'  25"  =  8.253180  =  cos  C. 

log  sin  a  =  86°  51'  =  9.999343 
log  sin  b  =  18°  3'  32"  =  9^491354 
b  —  10      =  18°    1'  50"  =  9.490697  =  sin  b. 

log  cos  a  =  86°  51'  =  8.739969 
log  cos  b  =  18°  1'  50"  =  9.978143 
c  +  10      =  86°  41'  14"  =  9.761826  =  cos  c. 

2.  Given  b  =  155°  27'  54",  and  c  =  29°  46'  8",  to  find  a,  B, 
and  C. 

Ans.  a  =  142°  9'  13",  B  =  137°  24'  21",  and  C  =  54°  1'  16". 

3.  Given  B  =  47°  13'  43",  and  C  =  126°  40'  24",  to  find  a, 
b,  and  c. 

Ans.  a  =  133°  32'  26",  b  =  32°  8'  56",  and  c  =  144°  27'  3". 

Eem. — As  the  formulas  are  constructed  with  unity  as  radius, 
if  logarithms  are  used,  when  the  formula  is  a  product,  10  must 
be  subtracted,  but  when  a  quotient,  10  must  be  added. 

A  spherical  triangle  which  has  one  of  its  sides  a  quadrant, 
is  called  a  Quadrantal  Triangle,  and  is  readily  solved  by  passing 
to  its  polar  triangle,  which  will  be  right-angled,  solving  it,  and 
returning  to  the  quadrantal  triangle. 

The  supplement  of  any  side  of  a  triangle  is  equal  to  the  oppo- 
site angle  of  the  polar  triangle,  and  the  supplement  of  any  angle 
is  equal  to  the  opposite  side  of  the  polar  triangle.  The  return  is 
effected  in  the  same  way  as  each  triangle  is  polar  to  the  other. 


SPHERICAL     TRIGONOMETRY, 


109 


PKOBLEM    I. 

To  show  that  the  sines  of  the  sides  of  a  spherical  trian- 
gle are  respectively  proportional  to  their  opposite  angles. 

Let  ABC  be  any  oblique-angled 
triangle. 

From  either  vertex,  as  A,  draw 
an  arc  of  a  great  circle  perpendicu- 
lar to  the  opposite  side ;  then  will 
the  triangles  ABD  and  ADCbe  right- 
angled  at  D,  and 

sin  V  =  sin  c  x  sin  B  ; 


and 


sin  V  =  sin  b  x  sin  C. 


and 


sin  b  x  sin  C  =  sin  c  x  sin  B; 
sin  b  :  sin  c  : :   sin  B  :  sin  C. 


In  like  manner,    sin  a  :  sin  b  : :  sin  A  :  sin  B  ; 
and  sin  a  :  sin  c  : :   sin  A  :  sin  C. 


The  result  is  the  same  when  the  perpendicular  falls  on  the 
opposite  side  produced. 

In  the  triangle  ABD  and  in  the 
triangle  ACD, 

sin  V  =  sin  c  x  sin  B ; 

sin  V  =  sin  b  x  sin  C. 

.*.     sin  b  x  sin  C  =  sin  c  x  sin  B ; 


and 


sin  b  :  sin  c  : :  sin  B  :  sin  C,  etc. 


110 


SPHERICAL     TRIGONOMETRY. 


PROBLEM    II. 

In  an  oblique-angled  spherical  triangle,  if  from  the 
vertex  of  either  angle  an  arc  be  drawn  perpendicular  to 
the  opposite  side,  dividing  it  into  two  segments,  find  these 
segments. 

Let  ABC  be  any  oblique-angled  spher- 
ical triangle. 

From  either  vertex,  as  C,  draw  CD  an 
arc  of  a  great  circle  perpendicular  to  the 
opposite  side  ;  then,  from  5th  formula  of 
Napier,  (s  -f-  8*  =;  c). 

In  the  triangle  ACD  and  in  the 
triangle  BCD, 

cos  b 


cos  a  =  cos  p  x  cos 
cos  b       cos  p  x  cos  s' 


and 


cos« 
cos  a 


cos  p  x  cos  s 
cos  b   : :   cos  s 


cos  p  x  cos  s  ; 

_  cos  s' 
~  cos  s  ' 
cos  s' : 


and  by  composition  and  division, 
cos  a  —  cos  b  :  cos  a  -f  cos  b  : : 
cos  a  —  cos  b 


cos  s  —  cos  s'  :  cos  s  -f  cos  s'. 
cos  s  —  cos  s' 


COS 


-|-  cos  s'  * 


cos  a  +  cos  b 
and  from  Prob  5,  Plane  Trig., 

cos  M  —  cos  N  sin  -|  (M  -j-  N)  x  sin  -J-  (M  —  N) 


cos  M  -f-  cos  N 


cos  a  —  cos  b 
cos  a  -f-  cos  b 


cos i  (M  +  N)  x  cos  J  (M  —  N) 
=  —  tang.  J(M  +  N)x  tang.  J  (M  -  N). 

=  —  tang.  I  (a  +  b)  x  tang.  £  (a  —  b) ; 


cos  5  —  cos  s 


-,  =  —  tang.  £  (s  +  s1)  x  tang.  J  (5  —  s'). 


and 

cos  s  -+-  cos  s 

And 

tang.  £  (5  +  «')  x  tang.  £  (s—s')  —  tang.  £  («  +  b)  x  tang.  J  (a— b)  ; 

.*.  tang  £  (s  -f-  5') :  tang.  £  {a  -\-  b)  ::  tang.  i(a— b)  :  tang.  £  (s— »'). 

KBIT. — 5  and    *'  being    determined,  in    each    right-angled 
triangle  are  known  two  sides  and  an  angle  opposite  one  of  them. 


SPHERICAL     TEIGONOMETEY, 


111 


PROBLEM    III. 

When  two  sides  and  the  included  angle  are  given,  to 
find  the  other  parts. 

Let  ABC  be  an  oblique  spherical  tri- 
angle, a,  c,  and  B  given. 

From  A  draw  AD  an  arc  of  a  great 
circle  perpendicular  to  the  opposite 
side  BC,  and  in  the  triangle  ABD, 


and 
and 


sin^  =  sin  c  x  sin  B; 

.     _APk       cos  B 

sin  BAD  = : 


aD 


sin  BD  =  sin  c  x  sin  BAD. 
DC  =  a  —  BD; 
cos  o  =  cos  p  x  cos  DC  ; 


and 


sin  C  == 


sin  CAD 


sin  p  m 
sin  b9 
sin  DC 


sin  b 
angle  A  s=  angle  BAD  +  angle  CAD ; 

hence,  b,  A,  and  C  are  determined. 


PROBLEM    IY. 

When  a  side  and  the  two  adjacent  angles  are  given. 

Let  B,  C,  and  a  be  given ;  then  in 
the  polar  triangle, 


and 


I  =  180  —  B, 
c  —  180  —  c, 
A  =  180  — «; 


that  is,  two  sides  and  the  included  angle 
known.  Solve  the  polar  triangle  by 
Problem  3,  and  return  to  the  original  triangle 


112 


SPHERICAL     TRIGONOMETRY. 


PKOBLEM    V. 

Wlien  two  sides  and  an  angle  opposite  one  of  them  is 
given. 

Let  B,  b,  and  c  be  given,  and  from 
the  angle  A,  opposite  the  unknown 
side  a,  draw  an  arc  of  a  great  circle 
perpendicular  to  it. 

In  the  triangle  ABD, 

sin  jp  =  sin  c  x  sin  B, 

.     DAr.        cos  B 

sin  BAD  = , 

cosp 

and    sin  BD  =  sin  c  x  sin  BAD. 


In  the  triangle  ADC, 


and 


cos  DC  = 


sin  CAD  = 


cos  b 
cos  p' 
sin  DC 


sin  b   9 
angle  A  =  angle  BAD  +  angle  CAD ; 

hence  all  the  parts  are  determined. 

Eem. — When  the  three  sides  are  given,  the  angles  are  found 
by  this  problem,  after  having  found  the  segments  of  one  side  by 
Problem  2. 


PEOBLEM    VI. 

When  two  angles  and  a  side  opposite  one  of  them  is 
given. 

Let  B,  C,  and  c  be  given;  then,  in  the 
polar  triangle, 

b  =  180  -  B, 

c  =  180  -  C, 

and         C  =  180  —  c; 

the  same  as  in  Problem  5,  and  must  be 
solved  accordingly,  and  then  return  to  the  original  triangle. 


SPHERICAL     TRIGONOMETRY.  113 

PROBLEM    VII. 

To  find  the  area  of  a  spherical  polygon. 

When  the  angles  are  not  given,  find  them  by  the  foregoing 
problems ;  then  from  Geometry,  Book  8, 

The  area  of  a  spherical  triangle  is  equal  to  the  product  of  its 
spherical  excess  and  the  trirectangular  triangle,  and  the  same  for 
any  polygon, 

EXAMPLES. 

1.  What  is  the  area  of  a  spherical  triangle  on  the  surface  of  a 
sphere  whose  diameter  is  20  feet ;  the  angles  of  the  triangle  arc 
A  =  130° ;  B  =  110°  ;  and  C  =  165°. 

130  .         ...      ,         ,     ,.       .         Wx  3.1416 

i  surface  of  trirectangular  triangle  =  5 

110  o 

165  =  157.08 

405  157.08  x|  =  392.7  sq.  ft.,  Ans. 

180 

-—  =s  f,  spherical  excess. 

2.  What  is  the  area  of  a  spherical  polygon  of  five  sides  on  a 
sphere  whose  diameter  is  40  feet,  and  the  sum  of  the  angles  of 
the  polygon  is  660°. 

402  x  3.1416       4 

g x  g  =  40  x  5  x  1.0472 

=  209.44  sq.  ft.,  Ans. 

3.  Find  the  area  of  a  spherical  polygon  of  eight  sides,  on  a 
sphere  30  feet  in  diameter,  and  each  angle  of  the  polygon  being 
150  degrees. 

302x3.1416       4       _.       ln      1KfV_Q 
g —  X  o  =  30  x  10  x  1.5708 

=  471.24  sq.  ft.,  Ans. 


in 


SPHERICAL     TRIGONOMETRY. 


PROBLEM    VIII. 

To  find  the  shortest  distance,  on  the  surface  of  the  earth, 
between  two  places  whose  latitudes  and  longitudes  are 
known. 

Rem.— The  shortest  distance  between  two  points  on  the  sur- 
face of  the  earth  is  measured  on  the  arc  of  a  great  circle  joining 
the  points. 

EXAMPLES. 

1.  The  latitude  of  New  York  City  is  40°  48' ;  its  longitude  3° 
east;  the  latitude  of  San  Francisco  is  37°  45'  north,  and  its 
longitude  45°  40'  west.     What  is  the  distance  between  them  ? 

The  radius  of  the  earth  is  39G2  miles,  making  69.15  miles  to 
a  degree.  Ans.  37°  18'  46"  ==  2580.18  miles. 

Let  this  figure  represent  a  hemi- 
sphere; NS  a  meridian  passing  through 
Washington ;  EQ,  equator.  The  point 
C  represents  New  York,  and  B'  San 
Francisco ;  the  point  B  is  at  the  North 
Pole;  BC  and  BB'  are  the  colatitudes  of 
New  York  and  San  Francisco,  and  the 
angle  B  the  difference  of  longitude  of  C 
and  B'. 

From  C  draw  CA  perpendicular  to  BB' ;  then  in  the  triangle 
BB'C,  angle  B  =  48°  40',  the  side  a  =  49°  12',  and  BB'  ==  52°  15; 

and  as  sin  b  =  sin  a  x  sin  B  =  34°  38'  23" 


and 


cos  c 


37°  25'  14" 


cos  a  _ 
cos  b 

c  =&  52°  15'  —  37°  25'  14"  =  14°  49'  46" ; 
and      cos  a'  —  cos  b  x  cos  c'  =  37°  18'  46"  =  2580.18  miles. 


2.  The  latitude  and  longitude  of  New  York  given,  also  the 
distance  from  New  York  to  San  Francisco,  and  the  latitude  of 
the  latter  place,  to  find  its  longitude, 

3.  Given  the  latitude  and  longitude  of  New  York,  the  distance 
to  San  Francisco  and  its  longitude,  to  find  the  latitude. 

Rem. — The  student  will  become  more  familiar  with  the  prin- 
ciples by  finding  the  different  parts  of  the  same  problem,  than  by 
taking  different  orres. 


OBRAfl 


ok  xms 


XJNIVERSITY 


SPHERICAL     TRIGON 


115 


PROBLEM    IX. 

To  -find  the  hour  of  the  day ;  the  altitude  of  the  sun,  its 
declination  and  the  latitude  of  the  observer  being  given. 

The  spherical  triangle  of  which  we  know  the  three  sides  are 
in  the  celestial  concave.  Its  vertices  are  the  sun,  the  zenith  of 
the  observer,  and  the  Celestial  Pole,  or  the  point  in  the  heavens 
pierced  by  the  axis  of  the  earth,  perpendicular  to  the  equator. 

The  arc  of  the  great  circle  joining  the  sun  and  the  pole  is  the 
codeclination  of  the  sun,  when  the  sun  and  the  observer  are  both 
on  the  same  side  of  the  equator;  when  they  are  on  different 
sides  of  the  equator,  it  is  the  sum  of  the  declination  and  90°. 

The  Coaltitude  of  the  sun  is  the  arc  of  the  great  circle 
joining  the  sun  and  the  zenith  of  the  observer;  and  the 

Colatitude  of  the  observer  is  the  arc  joining  the  zenith 
and  the  pole. 


zc 


EXAMPLE. 

In  latitude  36°  40'  the  declination  of  the  sun  is  12°  20'  N., 
and  its  altitude  30°  30'.     What  is  the  hour  of  the  day  ? 

Ans.  Either  7h.  56m.     2 sec.  a.m., 
or  4  h.     3  m.  58  sec.  p.  m. 

In  this  example  the  three  sides  are  given  to  find  the  angle  at 
the  pole,  which  is  the  hour  angle,  and 
being  reduced  from  degrees,  etc.,  to 
hours,  minutes,  etc.,  by  dividing  by  15, 
gives  either  the  time  before  or  after 
12  m.  The  angle  having  its  vertex  at 
the  pole,  one  side  of  which  extends  from 
the  pole  to  the  sun,  the  other  to  the 
zenith. 


The  sun,  the  zenith,  and  the  celes- 
tial pole  N  are  the  vertices  of  the  triangle 

BCB' ;  the  three  sides  are  given.  Draw  CA  perpendicular  to 
BB' ;  then  find  the  segments  of  c,  8,  and  tf;  and  then  the  angle 
B,  which  reduce  to  hours,  etc.,  and  it  is  either  so  long  before  or 
after  12  o'clock  m. 


116  SPHERICAL     TRIGONOMETRY. 


PROBLEM    X. 

To  find  the  length  of  the  day  at  any  place,  the  latitude 
and  declination  of  the  sun  being  known. 

Let  NS  be  the  meridian  at  which 
the  sun  reaches  the  horizon  when  it  is 
on  the  equator ;  that  is,  when  it  rises  at 
6  o'clock.  When  the  sun  has  a  declina- 
tion north,  it  will  be  at  s  on  the  ecliptic, 
instead  of  being  at  C  on  the  same  meri- 
dian at  6  o'clock. 

It  has  already  passed  the  distance 
Bs  above   the   horizon,   and  the  time 

taken  for  this  passage  is  in.  the  same  proportion  to  24  hours,  that 

this  arc  AC  is  to  360  degrees.     The  angle  is  AN C,  and  is  measured 

by  the  arc  AC. 

In  the  triangle  ABC,  right  angled  at  A,  AB  is  equal  to  the 

declination  of  the  sun  ;  the  angle  ACB  =  ECH  is  the  coaltitude 

of  the  place. 

EXAM  PLE. 

What  is  the  length  of  the  day  in  latitude  40°  30'  north,  when 
the  declination  of  the  sun  is  12°  50'  ? 

T  will  be  the  position  of  the  traveler. 

sin  l  =  cot  C  x  fcang.  c ; 

.\  log  tang.  comp.  C  40°  30'  =  9.931499 
log  tang,  c  -  10  12°  50'  =  9.357566 
sin  b  =  sin  C        11°  13'  10"       9.289065 

Time  before  6  o'clock  that  the  sun  rises  and  of  course  the 
same  time  after  6  it  sets. 

h.    min.     sec. 

11°  13'  10"  =  0    44    53 
2 


Hence,  1    29    46 

12 0 0 

Length  of  day,  13    29    46 


SPHERICAL     TRIGONOMETRY.  117 

Twice  the  time  of  the  sun  passing  from  the  horizon  to  the 
meridian  NS  must  be  added  to  12  hours  to  get  the  length 
of  the  day. 

Rem. — As  a  traveler  goes  north,  starting  at  the  equator,  for 
every  degree  that  he  travels,  the  south  pole  recedes  one  degree ; 
therefore,  the  angle  HCS  measures  his  latitude,  and  HCE  is  his 
colatitude.  This  is  the  same  as  the  north  pole  rising  a  degree 
for  every  degree  he  travels  ;  hence,  the  altitude  of  the  north  pole 
is  his  latitude. 


TA  B  L  E, 


1<> 


COMTAINI.VO 


THE  LOGARITHMS  OF  NUMBERS 


FROM    1    TO    10,000. 


Hf 


6> 


n 


it 


NUMBERS  FROM  1  TO  100  AND  THEIR  LOGARITHMS, 
WITH  THEIR  INDICES. 

tu0« 

»  9 

[   No. 

Logarithm. 

Na 

Logarithm. 

No. 

Logarithm. 

No. 

Logarithm. 

No. 

Loga  ithm. 

!1- 

0000000 

21 

1-322219 

41 

1-612784 

61 

1-785330 

81 

1908485  i 

0-301030 

22 

1-342423 

42 

1-623249 

62 

1-792392 

82 

1-913814 

3 

0  477121 

23 

1-361728 

43 

1-633468 

63 

1*799341 

83 

1-919D78 
1-924279 

)  4 

0*003060 

24 

1  -3802 11 

44 

1-643453 

64 

1-800180 

84 

.*«* 

1  5 

0-698970 

25 

1-397940 

45 

1-653-213 

65 

1-812913 

85 

1-929419 
1-934498 

>  6 

0778151 

26 

1-414973 

46 

1-662V58* 

66 

1819544 

86 

(  7 

Q-845M8 

27 

1-431304 

47 

1-672098 

67 

1-826075 

87 

1-939519  / 

• 

?  8. 

0-903090 

28 

1-447158 

48 

1-681241 

68 

1-832509 

88 

1-944483 

(  9 

0  954243 

29 

1-462398 

49 

1-690 193 

69 

1-838849 

89 

1-949390 
1-954243 

1-959041 

i  10 

1-000000 

30 

1-477121 

50 

1-698970 

7J 

1-845098 

90 

)  U 

1041393 

31 

1-491362 

51 

1-707570 

71 

1851258 

91 

)  12 

1079181 

32 

1-505150 

52 

1-716003 

72 

1-857332 

92 

1-963788 

)  13 

1113943 

33 

1  518.314 

53 

1-724276 

73 

1-863323 

93 

1-968483 

;  W 

1  140128 

34 

1-531479 

54 

1*732394 

74 

1669232 

94 

1973128 

i  15 

1176091 

35 

1-544068 

55 

1-740363 

75 

1.875061 

95 

1-977724 

I  w 

1-204120 

36 

1-556303 

56 

1-748188 

76 

1-880814 

96 

1-982271 

17 

1-2:10449 

37 

1-568202 

57 

1-755875 

77 

1-886491 

97 

1-986772 

(  18 

1-255273 

38 

1-579784 

58 

1-763428 

78 

1-892095 

98 

1-991226 

(  I9 

1-278754 

39 

1-591065 

59 

1-770852 

79 

1-897627 

99 

1-995635 

>  20 

1-301030 

40 

1-602060 

60 

1-778151 

80 

1-903090 

100 

2000000 

t*  J  * 


Note. — In  the  following  part  of  the  Table,  the  Indices  are  omitted,  as  they 
can  be  very  easily  supplied.  Thus,  the  index  of  the  logarithm  of  every  integer 
number,  consisting  only  of  one  number,  is  0;  of  two  figures,  1;  of  three 
figures,  2;  of  four  figures,  3:  being  always  a  unit  less  than  the  number  of 
figures  contained  in  the  integer  number.  The  index  to  the  logarithm  of  every 
number  above  100,  in  the  following  part  of  the  Table,  is  omitted  ;  yet,  in  the 
operation,  it  must  be  prefixed,  according  to  this  remark 


of  600  is  2-77815,  and  that  of  6000  is  377815, 


so  that  the  logarithm 
ind  so  of  the  rest. 


ISO 


LOGARITHM 


No.  |      0       |      1       |      3      |      3      |      4:      | 


I      6      | 


100 

000000 

1 

4321 

s 

8600 

3 

012837 

4 

7033 

s 

021189 

G 

5306 

7 

9384 

8 

033424 

9 

7426 

110 

041393 

1 

5323 

2 

9218 

3 

053078 

4 

6905 

5 

060698 

0 

4458 

7 

8186 

8 

071882 

0 

5547 

120 

079181 

1 

082785 

2 

6360 

3 

9905 

41093422 

5 

6910 

6 

100371 

7 

3804 

8 

7210 

9 

110590 

130 

113943 

1 

7271 

2 

120574 

3 

3852 

4 

7105 

5 

130334 

6 

3539 

7 

6721 

8 

9879 

9 

143015 

140 

146128 

1 

9219 

1 

152288 

3 

5336 

4 

8362 

5 

161368 

f) 

4353 

7 

7317 

B 

170262 

9 

3186 

ISO 

176091 

1 

8977 

2 

181844 

3 

4691 

4 

7521 

5 

190332 

6 

3125 

7 

5900 

8!  8657 

9 

201397 

000434 
4751 
9026 

013259 
7451 

021603 
5715 


041787 
5714 
9606 

053463 
7286 

061075 
4832 
8557 

072250 
5912 

079543 

083144 
6716 

090258 
3772 
7257 

100715 
4146 
7549 

110926 

114277 
7603 

120903 
4178 
7429 

130655 
3858 
7037 

140194 
3327 

146438 
9527 

152594 
5640 
8664 

161667 
4650 
7613 

170555 
3478 

176381 
9264 

182129 
4975 
7803 

190612 
3403 
6176 
8932 

201670 


000868 
5181 
9451 

013680 
7868 

022016 
6125 

030195 
4227 
8223 

042182 
6105 
9993 

053846 
7666 

061452 
5206 


079904 

083503 
7071 

090611 
4122 
7604 

101059 
4487 
7888 

111263 

114611 
7934 

121231 
4504 
7753 

130977 
4177 
7354 

140508 


176670 
9552 

182415 
5259 
8084 

190892 
3681 
6453 
9206 

201943 


001301 
5609 
9876 

014100 
8284 

0224^8 
6533 

030600 
4628 
8620 

042576 
6495 

050380 
4230 
8046 

061829 
5580 
9298 

072985 
6640 

080266 
3861 
7426 

090963 
4471 
7951 

101403 


114944 
8265 

121560 
4830 
8076 

131298 
4496 
7671 

140822 
3951 

147058 

150142 
3205 
6246 
9266 

162266 
5244 
8203 

17114! 
4060 

176959 
9839 

182700 
5542 
8366 

191171 
35)51) 
6729 
9481 

202216 


001734 
6038 

010300 
4521 
8700 

022841 
6942 

031004 
5029! 
9017 


002166  ( 
6466 

010724  ( 
4940! 
9116! 

023252  ( 
7350 1 

031408  ( 
5430 
9414 

043362 

i  7275 

051153 

4996 


043755 
766-1 

051538 
5378 
9185 

062358 


080626 
4219 
7781 

091315 
4820 
8298 

101747 
5169 
8565 

111934 

115278 

8595 

121888 

5156 

8399 

131619 

4814 

7987 

141136 

4263 

147367 

150449 
3510 
6549 
P567 

162164 
5541 
8497 

171434 
4351 

177248 

180126 
2985 
5825 
8647 

191451 
4237 
7005 
9755 

202488 


080987 
4576 
8136 

091667 
5169 
8644 

102091 
5510 
8903 

112270 


131939 
5133 
8303 

141450 
4574 

147676 

150756 

3815 

6852 


177536 

180413 
3270 
6108 
8928 

191730 
4514 
7281 

200029 
2761 


081347 
4934 
8490 

092018 
5518 
8990 

102434 
5851 
9241 

112605 

115943 
9256 

122544 
5806 
9045 

132260 
5451 
8618 

141763 
4885 


t 

8  | 

003029 

003461 

7321 

7748 

011570 

011993 

5779 

6197 

9947 

020361 

024075 

4486 

8164 

8571 

032216 

032619 

6230 

6629 

040207 

040602 

044148 

044540 

8053 

8442 

051924 

052309 

5760 

6142 

9563 

9942 

063333 

063709 

7071 

7443 

070776 

071145 

4451 

4816 

8094 

8457 

081707 

082067 

5291 

5647 

8845 

9198 

092370 

092721 

5866 

6215 

9335 

9681 

102777 

103119 

6191 

6531 

9579 

9916 

112940 

113275 

116276 

11G608 

9586 

9915 

122871 

123198 

6131 

6456 

9368 

9690 

132580 

132900 

5769 

6086 

8934 

9249 

142076 

142389 

5196 

5507 

148294 

148603 

151370 

151676 

4424 

4728 

7457 

7759 

1604(59 

160769 

3460 

3758 

6430 

6726 

9380 

•  9674 

172311 

172603 

5222 

5512 

178113 

178401 

180986 

181272 

3839 

4123 

6674 

6956 

9490 

9771 

192289 

192567 

5069 

5346 

7832 

8107 

200577 

200850 

3305 

3577 

9    |  DiffA 


8978 
033021 

7028 
040998 

044932  393  ; 
8830  390  > 

052694 1 386 
6524 i 383 

0603201379 
40831376 
7815 !  373 

0715141370 
5182J366 
8819 i 363 


1300121323 


I  0  |  1 


I  7     |  8  |  9  |D,ff. 


OF     NUMBERS. 


121 


no.  ;  o  |  1  |  a  T^^M^sn^sn^ls^TonTuiffl 

160 1 204120  204391 

204063 

204934 

205204  205475  205746 

206016 

206286 1 206556 | 271  \ 

1 

0826 

7096 

7365 

7634 

7904 

8173 

8441 

8710 

8979   9247 

269  ) 

2 

9515 

9783 

210051 

210319 

210586 

210853 

211121 

211388 

211654 

211921 

267) 

3 

212188 

212454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266  S 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264  S 

5 

7484 

7747 

8010 

8273 

8536 

.  8798 

9060 

9323 

9585 

9840 

262  ) 

6 

220108 

220370 

220631 

220892 

221153 

221414 

221675 

221936 

222196 

222450 

261  ( 

7 

27  lu 

2976 

3230 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259  ( 

8 

5309 

5568 

5826 

6084 

6342 

6000 

6858 

7115 

7372 

7630 

258  ( 

>   9 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

230193 

256  J 

•170 

230449 

230704 

230960 

231215 

231470 

231724 

231979 

232234 

232488 

232742 

255; 

'   1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253  > 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252) 

3 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

240050 

240300 

250) 

4 

240549 

240799 

241048 

241297 

241546 

241795 

242044 

212293 

2541 

2790 

249  ) 

5 

3038 

3280 

3534 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248) 

>   6 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246) 

>   7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9087 

9932 

250176 

245) 

>   8 

250420 

250664 

250908 

251151 

251395 

251638 

251881 

252125 

252368 

2610 

243  > 

>   9 

2853 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242  j 

>  183 

255273 

255514 

255755 

255996 

256237 

256477 

256718 

256958 

257198 

257439 

241  > 

1 

7079 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239.' 

2 

200071 

260310 

260548 

260787 

261025 

261263 

261501 

261739 

261976 

262214 

238? 

3 

2451 

2688 

2925 

3102 

3399 

3636 

3873 

4109 

4346 

4582 

237? 

'   4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235  ( 

1   5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234) 

>   6 

9513 

9746 

9980 

270213 

270446 

270679 

270912 

271144 

271377 

271609 

233) 

i   7 

271842 

272074 

272306 

2538 

2770 

3001 

3233 

3464 

3696 

3927 

232) 

f   8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230  > 

1   9 

6402 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

8525 

229) 

!  190 

278754 

278982 

279211 

279439 

279667 

279895 

280123 

280351 

280578 

280806 

228{ 

1 

281033 

281261 

281488 

281715 

281942 

282169 

2396 

2622 

2849 

3075 

227  ( 

1   2 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226  ( 

3 

5557 

5782 

6007 

6232 

G456 

6681 

6905 

7130 

7354 

7578 

225  ( 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

9539 

9812 

223  ( 

290035 

290257 

390480 

290702 

290925 

291147 

291369 

291591 

291813 

292034 

222  ( 

(   6 

2256 

2478 

2099 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221  ( 

(   7 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220/ 

8 

6665 

6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219) 
218? 

9 

8853 

9071 

9289 

9507 

9725 

9943 

300161 

300378 

300595 

300813 

>2O0 

301030 

301247 

301464 

301681 

301898 

302114 

302331 

302547 

302764 

302980 

217  ( 

}   1 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216  ( 

J   2 

5351 

5566 

5781 

5990 

6211 

6425 

6639 

6854 

7068 

7282 

215  ( 

)   3 

7496 

7710 

7924 

8137 

8351 

8564 

8778 

8991 

9204 

9417 

213  < 

)   4 

9630 

9843 

310050 

310268 

310481 

310093 

310906 

311118 

311330 

311542 

212  ( 

>   5 

311754 

311966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

21H 

)   6 

3807 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210  < 

)   7 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209? 

5  8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208  i 

)   9 

320146 

320354 

320562 

320769 

320977 

321184 

321391 

321598 

321805 

322012 

207 
206  S 

\210 

322219 

322426 

322633 

322839 

323046 

323252 

323458 

323665 

323871 

324077 

{      1 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205  { 

\      2 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204  ( 

S      3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

330008 

330211 

203  ( 

S   4 

330414 

330617 

330819 

331022 

331225 

331427 

331630 

331832 

2034 

2236 

202  ( 

{      5 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202  ( 

(   6 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201  ( 

<   7 

6460 

6660 

6860 

7060 

7280 

7459 

7659 

7858 

8058 

82571200  ( 

J   8 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

340047 

3402461199  ( 
2225J198J 

9 

34044^ 

340642 

340841 

341039 

341237 

341435 

341632 

341830 

2028 

SiJLSU, 


j^zjl* 


122 


LOGARITHMS 


Na| 

o    | 

1  I 

«   1 

3  I 

4 

5   ! 

6  | 

7    ! 

s    ; 

9 

DiS. 

(  220 

342423 

342620  342817 

343014 

343212 

343409 

343606 

343802 

343999 

344196, 197  I 

(   1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

59(52 

6157  196  ( 

S   2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110  195  ( 

(   3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

98M 

350054  194  ( 
1989  193  ( 
3916  193  { 
5834  192/ 

(      4 

350248 

350442 

350636 

350829 

351023 

351216 

351410 

351603 

351796 

S   5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

(   6 

4108 

4301 

4493 

46851  4876 

5068 

5260 

5452 

5643 

i   7 

\      8 

6026 

6217 

6408 

65991  6790 

6981 

7172 

7363 

7554 

7744  191  / 

7935 

8125 

8316 

8506  8696 

8886 

9076 

9266 

9456 

9646 

190 : 

189  ( 
188) 

)   9 

9835 

360025 

360215 

360404 

360593 

360783 

360972 

361161 

361350 

361539 

}230 

361728 

361917 

362105 

362294 

362482 

362671 

362859 

363048 

363236 

363424 

J   1 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188, 

)   2 

5488 

5675 

5862 

6049 

6236 

6423 

6610 

6796 

6983 

7169 

187) 

)   3 

7356 

7542 

7729 

7915 

8101 

8287 

8473 

8659 

8845 

9030  186  ) 

;  4 

9216 

9401 

9587 

9772 

9958 

370143 

370328 

370513 

370698 

370883  J 85  ) 

5  5 

371068 

371253 

371437 

371622 

371806 

1991 

2175 

2360 

2544 

2728 

184  ) 

)  6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184) 

)   7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183) 

)   8 

6577 

6759 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182) 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

380030 

181  S 

)S40 

380211 

380392 

380573 

380754 

380934 

381115 

381296 

381476 

381656 

381837 

181  \ 

(      1 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180  ( 

/   2 

3815 

3995 

4174 

4353 

4533 

4712 

4891 

5070 

5249 

5428 

179  f 

(   3 

5606 

5785 

5964 

6142 

6321 

6499 

6077 

6856 

7034 

7212 

178  < 

1   ^ 

7390 

7568 

7746 

71*3 

8101 

8279 

8456 

8634 

8811 

8989 

178  ( 

I      5 

9166 

9343 

9520 

9698 

9875 

390051 

390228 

390405 

390582 

390759 

177  ( 

?   6 

390935 

391112 

391288 

391464 

391041 

1817 

1993 

2169 

2345 

2521 

176  ( 

1   7 

2697 

2873 

3048 

3224 

3400 

3575 

3751 

3926 

4101 

4277 

176/ 

1   8 

4452 

4627 

4802 

4977 

5152 

5326 

5501 

5676 

5850 

6025 

175  [ 

1   9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174  J 

(  250 

397940 

398114 

398287 

398461 

398634 

398808 

398981 

399154 

399328 

399501 

173) 

(   1 

9674 

9847 

400020 

400192 

400305 

400538 

400711 

400883 

401056 

401228 

173) 

S   2 

401401 

401573 

1745 

1917 

2JS9 

2201 

2433 

2605 

2777 

2949 

172) 

S   3 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

171) 

S   4 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171) 

S   5 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

170) 

(   6 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

169/ 

(      7 

9933 

410102 

410271 

410440 

410009 

410777 

410946 

411114 

411283 

411451 

169> 

\      8 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2954 

3132 

168 
167  < 

>   9 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4039 

4806 

)  260 

414973 

415140 

415307 

415474 

415641 

415808 

415974 

416141 

416308 

416474 

167/ 

)   1 

6641 

6807 

6973 

7139 

7300 

7472 

7638 

7804 

7970 

8135 

166/ 

)      2 

8301 

8467 

8633 

8798 

8964 

9129 

9295 

9460 

9625 

9791 

165/ 

;    3 

9956 

420121 

430386 

420451 

420016 

420781 

420945 

421110 

421275 

421439 

165/ 

)   4 

421604 

1768 

1933 

2097 

2261 

2426 

2590 

2754 

2918 

3082 

164/ 

)   5 

3246 

3410 

3574 

3737 

3901 

4005 

4228 

4392 

4555 

4718 

164/ 

)   6 

4882 

5045 

5208 

5371 

5531 

5697 

5860 

6023 

6186 

6349 

163/ 
162/ 

)   ? 

6511 

6674 

6836 

6999 

7161 

7324 

7486 

7648 

7811 

7973 

)   8 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162/ 

(   9 

9752 

9914 

430075 

430230 

430398 

430559 

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431203 

161  > 

?  270 

431364 

431525 

431685 

431846 

432007 

432167 

432328 

432488 

432649 

432809 

161  I 

/   1 

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3130 

3290 

3450 

3610 

3770 

3930 

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4249 

4409 

100  \ 

?   2 

4569 

4729 

4888 

5048 

5207 

5367 

5526 

5685 

5844 

6004  159  \ 

(   3 

6163 

6322 

6481 

6640 

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7116 

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0175  158  ( 

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7751 

7909 

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8701 

8859 

9017 

(   5 

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440437 

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/   8   4043 
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5449  156 
7003  155 

5760 

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6537  6692 

6848 

>  Ntk 

1  o 

11 

I  iU^J.  ~5L>ULJ^LXJL>L^~L2vJ  "^ 

OP     NUMBERS. 


123 


If  a  |  0  |   1  |  %     |  3 

1  * 

5 

6 

7 

8 

1  9 

DiS 

2801447158 

447313  447468 

447623 

447778  447933 

448068 

448242 

448397 

448552;  155 

l|  8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

J94J 

450095  154 

2  450249 

450403 

450.557 

450711 

450865 

451018 

451172 

451326 

451479 

1633  154 

3 

1786 

1940 

2093 

2247 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

5302 

5454 

5606 

5753 

5910 

6062 

6214 

152 

1 

6366 

6518 

6670 

6821 

6973 

7125 

7276 

7428 

7579 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

8 

9392 

9543 

9694 

9845 

9995 

460146 

460296 

460447 

460597 

469748 

151 

9 

460898 

461048 

461198 

461348 

461499 

1649 

1799 

1948 

2098 

2248 

150 

290  462398 

462548 

462697 

462847 

462997 

463146 

463296 

463445 

463594 

463744 

150 

U  3893 

4042 

4191 

4340 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

6120 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

7312 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085 

9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

470116 

470263 

470410 

470557 

470704 

470851 

470998 

471145 

147 

6 

471292 

471438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

146 

7 

2756 

2903 

3049 

3195 

3341 

3487 

3633 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397 

6542 

6687 

6832 

6976 

145 

300 

477121 

477266 

477411 

477555 

477700 

477844 

477989 

478133 

478278 

478422 

145 

1 

8566 

8711 

8855 

8999 

9143 

9287 

9431 

9575 

9719 

9863 

144 

2 

480007 

480151 

480294 

480438 

480582 

480725 

480869 

481012 

481156 

481299 

144 

3 

1443 

1586 

1729 

1872 

2016 

2159 

2302 

2445 

2588 

2731 

143 

4 

2874 

3016 

3159 

3302 

3445 

3587 

3730 

3872 

4015 

4157 

143 

5 

4300 

4442 

4585 

4727 

4869 

5011 

5153 

5295 

5437 

5579 

142 

6 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

6997 

142 

7 

7138 

7280 

7421 

7563 

7704 

7845 

7986 

8127 

8269 

8410 

141 

8 

8551 

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8833 

8974 

9114 

9255 

9396 

9537 

9677 

9813 

141 

9 

9958 

490099 

490239 

490380 

490520 

490661 

490801 

490941 

491081 

491222 

140 

310 

491362 

491502 

491642 

491782 

491922 

492062 

492201 

492341 

492481 

492621 

140 

1 

2760 

2900 

3040 

3179 

3319 

3458 

3597 

3737 

3876 

4015 

139 

2 

4155 

4294 

4433 

4572 

4711 

4850 

4989 

5128 

5267 

5406 

139 

3 

5544 

5683 

5822 

5960 

6099 

6238 

6376 

6515 

6653 

6791 

139 

4 

6930 

7068 

7206 

7344 

7483 

7621 

7759 

7897 

8035 

8173 

138 

5 

8311 

8448 

8586 

8724 

8862 

8999 

9137 

9275 

9412 

9550 

138 

C 

9687 

9824 

9962  50009!) 

500236 

500374 

500511 

500648 

500785 

500922 

137 

7 

501059 

501196 

501333 

1470 

1607 

1744 

1880 

2017 

2154 

2291 

137 

8 

2427 

2564 

2700 

2837 

2973 

3109 

3246 

3382 

3518 

3655 

136 

B 

3791 

3927 

4063 

4199 

4335 

4471 

4607 

4743 

4878 

5014 

136 

320 

505150 

505286 

505421 

505557 

505693 

505828 

505964 

506099 

506234 

506370 

136 

1 

6505 

6640 

6776 

6911 

7046 

7181 

7316 

7451 

7586 

7721 

135 

2 

7856 

7991 

8126 

8260 

8395 

8530 

8664 

8799 

8934 

9068 

135 

3 

9203 

9337 

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9740 

9874 

510009 

510143 

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510411 

134 

4 

510545 

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510813 

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511081 

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1616 

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134 

5 

1883 

2017 

2151 

2284 

2418 

2551 

2684 

2818 

2951 

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133 

e 

3218 

3351 

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3617 

3750 

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133 

7 

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5476 

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5741 

133 

8 

5874 

6006 

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6535 

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132 

9 

7196 

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8119 

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330  518514 

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519040 

519171 

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519434 

519566 

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131 

Jl  9828 

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520090 

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520353 

520484 

520615 

520745 

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131 

2 

521138 

521269 

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1530 

1661 

1792 

1922 

2053 

2183 

2314 

131 

3 

2444 

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2705 

2835 

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3090 

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3486 

3616 

130 

4 

3746 

3876 

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4396 

4526 

4656 

4785 

4915 

130 

5 

5045 

5174 

5304 

5434 

5563 

5693 

5822 

5951 

6081 

6210 

129 

6 

6339 

6469 

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6727 

6856 

6985 

7114 

7243 

7372 

7501 

129 

7 

7630 

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8016 

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8402 

8531 

8660 

8788 

129 

8 

8917 

9045 

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9302 

9430 

9559 

9687 

9315 

9943 

530072  128 

9 

530200 

530328 

530456 

530584 

530712 

530840 

530968 

531096 

531223 

1351 1 128 

'No.  |      O 


a    I    3 


*   1  *   I   • 


7     |     8     |     0 


124 


LOGARITHMS 


rNo.|  0|1|2|3|4:|5|6|7|8  |  "fT"")^ 

1   340 

531479 

531807 

531734  531862 

531990 

5321171532245 

532372 

532500 

53C627 

128 

<   1 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

<   2 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

49  J  4 

5041 

5167 

127 

(   3 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

(   4 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126? 

/   5 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126  I 

I      6 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

540079 

540204 

125  I 

(   7 

540329 

540455 

540580 

540705 

540830 

540955 

541080 

541205 

1330 

1454 

125/ 

(   8 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125) 

9 

2825 

2950 

3074 

3199 

3323 

3447 

3571 

3696 

3820 

3944 

124) 

j  350 

544068 

544192 

544316 

544440 

544564 

544688 

544812 

544936 

545060 

545183 

124  i 

S   1 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124  < 

)   2 

6543 

6666 

678!) 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123  < 

)   3 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123  ( 

)   4 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

550106 

123  ( 

)   5 

550228 

550351 

550473 

550595 

550717 

550840 

550962 

551084 

551206 

1328 i  122  ( 

S   6 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122  ( 

<   7 

2668 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121  ( 

(   8 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121  ( 

(   9 

5094 

5215 

5336 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

121^ 

j  360 

556303 

556423 

556544 

556664 

556785 

556905 

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120  i 

)   1 

7507 

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7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120) 

i      2 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120) 

j   31  9907 

560026 

560146 

560265 

560385 

560504 

560624 

560743 

560863 

560982 

119$ 

>   4 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119) 

/   5 

2293 

2412 

2531 

2650 

2709 

2887 

3006 

3125 

3244 

3362 

119) 

/   6 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119) 

/    7 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

118  < 

)   8 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909!  118^ 

9 

70i>0 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

118  ( 

f  370 

568202 

568319 

568436 

568554 

568671 

568788 

568905 

569023 

569140 

569257 

117  > 

I      1 

9374 

9491 

9608 

9725 

9842 

9959 

570076 

570193 

570309 

570426 

117) 

?   2 

570543 

570660 

570776 

570893 

571010 

571126 

1243 

1359 

1476   1592 

117) 

/   3 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

116) 

l      4 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116) 

/   5 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116) 

h 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115) 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115) 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

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)   9 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

114  j 

(  380 

579784 

579898 

580012 

580126 

580241 

580355 

580469 

580583 

580697 

580811 

114/ 

<   1 

580925 

581039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114/ 

s    2 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

29:2 

3085 

114? 

3 

3199 

3312 

3426 

3539 

3052 

3765 

3879 

3992 

4105 

4218 

113) 

\    4 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113) 

<   5 

5461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113) 

I      6 

6587 

6700 

6812 

6925 

7037 

714J 

7262 

7374 

7486 

7599 

112) 

(   7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

112) 

<   8 

8832 

8944 

90.56 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

112) 

?   9 

9950 

590061 

590173 

590284 

590396 

590507 

590019 

590730 

590842  590953 

112  S 

)  390 

591065 

591176 

591287 

591399 

591510 

591621 

^91-32 

591843 

591955  592066 

111? 

)      1 

2177 

2288 

2399 

2510 

2621 

273-2 

2843 

2954 

3064 

3175,111/ 

C   o 

3286 

3397 

a508 

3618 

3729 

3840 

39.50 

4061 

4171 

4282  111  ) 

i   3 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

53861110  1 

)   4 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487  110) 

)   5 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

75861110) 

S      6 

7695 

7805 

7914 

8924 

8134 

8243  8353 

8462 

8572 

8681  110) 

$   7 

8791 

8900 

9009 

9119 

9228 

9337!  9446 

9556 

9665 

9774  109  ) 

(   8 

9883 

9992 

600101 

600210 

100319 

600428  600537 

600646 

600755  600864  109  S 

-9 

600973 

601082 

1191 

1299|  I4c8 

1517|  1625 

1734 

18431  1951  109 ^ 

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OP     NUMBERS. 


125 


9_Jj>ifnij 


126 


LOGARITHMS 


•  No.  |   O   | 


2  |  3   |  4 


I  7     j  8   |  9  | 


460:662758 


480 
1 
2 
3 
4 
5 
0 
7 
8 


3701 
4642 
558J 
6518 
7453 
8386 
9317 
670246 
1173 

672098 
3021 
3942 
4861 
5778 
6694 
7607 
8518 
9428 

680336 

681241 
2145 
3047 
3947 
4845 
5742 
6636 
7529 
8420 
9309 

690196 
1081 
1965 
284 
372' 
4605 
5482 
6356 
7229 
8101 


66J852 
3795 
4736 
5675 
6612 
7546 
8479 
9410 


510 


9838 
700704 
1568 
2431 
3291 
4151 
5008 
5864 
6718 

707570 
8421 
9270 

710117 
0963 
1807 
2650 
3491 
4330 
5167 


663041  663135  663230 


1265 

672190 
3113 
4034 
4953 

5870 
6785 
7698 
8609 
9519 
680426 

681332 
2235 
3137 
4037 
4935 
5831 
6726 
7618 
8509 
9398 

690285 
1170 
2053 
2935 
3815 
4693 
5569 
6444 
7317 
8188 

699057 
9924 

700790 
1654 
2517 
3377 
4236 
5094 
5949 
6803 

707655 
8506 
9355 

710202 
1048 
1892 
2734 
3575 
4414 
5251 


4*30 
5769 
6705 
7640 
8572 
9503 
670431 
1358 


3205 
4126 
5045 
5962 
6876 
7 
8700 
9610 
680517 

681422 
2326 
3227 
4127 
5025 
5921 
6815 
7707 
8598 
9486 

690373 
1258 
2142 
3023 
3903 
4781 
5657 
6531 
7404 
8275 

699144 
700011 
0877 
1741 
2603 
3463 
4322 
5179 
6035 
6888 

707740 
8591 
9440 

710287 
1132 
1976 
2818 
3659 
4497 
53:5 


3983 
4924 
5862 
6799 
7733 
8665 
9596 
670524 
1451 

672375 
3297 

4218 
5137 
6053 
6968 
7881 
8791 
9700 
680607 

681513 
2416 
3317 
4217 
5114 
6010 
6904 
7796 
8687 
9575 

690462 
134 
2230 
3111 
3991 
4868 
5744 
6618 
7491 
8362 

699231 
700098 
0963 
1827 
2689 
3549 
4408 
5265 
6120 
6974 

707826 
8676 
9524 

710371 
1217 
20(i0 
2902 
3742 
4581 
5418 


40781  4172 

5018  5112 

59561  6050 

6892 j  6986 

7826  7y20 

8759  8852 

9689  9782 
6706171670710 

1543  1636 


672467 
3390 
4310 

5228 
6145 
7059 
7972 
8882 
9791 


663324  663418 
4266 j  4360 
5206 


681603 
2506 
3407 
4307 
5204 
6100 
699-1 
7886 
8776 
9664 

690550 
1435 

2318 
3199 
4078 
4956 
5832 
6706 
7578 
8449 

699317 
700184 
1050 
1913 
2 
3635 
4494 
5350 
6206 
7059 

707911 
8761 
9609 

710456 
1301 
2144 
2986 
3826 
4665 
55i,2 


672560 
3482 
4402 
5320 
6230 
7151 
8063 
8973 
9882 

680789 

681693 
2596 
3497 
4396 
5294 
6189 
7083 
7975 
8865 
9753 

690639 
1524 
2406 
3287 
4166 
5044 
5919 
6793 
7665 
8535 

699404 
700271 
1136 
1999 
2861 
3721 
4579 
5436 
6291 
7144 

707996 
8846 
9694 

710540 
1385 
2229 
3070 
3910 
4749 
5586 


6143 
7079 
8013 
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9281 

9346 

9412 

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66  S 

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820070 

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820267 

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1186 

1251 

1317 

1382 

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1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

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2364 

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65  ( 

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2822 

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65  ( 

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4126 

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826464 

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65  J 

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65  > 

S   3 

8015 

8080 

8144 

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S   4 

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64  ) 

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64  ) 

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(   7 

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3147 

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63  ( 

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62  ( 

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62  ( 

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130 


LOGARITHMS 


No.  |      O      I      1      | 


I     3      |     4      |     5 


7     |     8      |     9      |Di£{ 


'700 

845098  845160 

845222 

845284 

845346 

845408 

845470 

845532 

845594 

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62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

62 

2 

(5337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

3 

C955 

7017 

7079 

7141 

7202 

7264 

732f 

7388 

7449 

7511 

62 

4 

7573 

7634 

7696 

7758 

7819 

78&1 

7943 

8004 

8066 

8128 

62 

'   5 

8189 

8251 

8312 

8374 

8435 

8559 

8620 

8682 

8743 

62 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

61 

7 

9419 

9181 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

'  8 

850033 

850095 

850156 

850217 

850279 

850340 

850401 

850462 

850524 

850585 

61 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

61 

710  851258 

851320 

851381 

851442 

851503 

851564 

851625 

851686 

851747 

851809 

61 

1   1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

o 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

61 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

61 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

61 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6008 

6668 

GO 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

60 

720 

857332 

857393 

857453 

857513 

857574 

857634 

857694 

857755 

857815 

857875 

60 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

ro 

2 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

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9018 

9078 

GO 

3 

9138 

9198 

9258 

9318 

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9439 

9499 

9559 

9619 

9679 

60 

4 

9739 

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9859 

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9978 

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860098 

860158 

860218 

8C0278 

60 

5 

860338 

860398 

8G0 158 

860518 

860578 

0037 

0697 

0757 

0817 

0877 

60 

6 

0937 

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1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

60 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

GO 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3144 

3204 

3263 

60 

720 

863323 

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8*53442 

863501 

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863G20 

863680 

863739 

863799  863858 

59  < 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59  < 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

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59  < 

3 

5104 

5163 

5222 

5282 

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54C0 

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5519 

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4 

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5814 

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6051 

6110 

6169 

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59 

5 

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6524 

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6701 

6760 

6819 

59 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

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59  i 

7 

7467 

7526 

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59  1 

8 

8056 

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8350 

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59  { 

9 

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8938 

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9056 

9114 

9173 

59  I 

740 

869232 

869290 

869349 

869408 

869466 

869525 

869584 

869642 

869701 

869760 

59  ! 

1 

9818 

9877 

9935 

9994 

870053 

970111 

870170 

870228 

870287 

8703-15 

59 

2 

S70404 

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0638 

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0813 

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0930 

58 

3 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

58 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

53  : 

58  ) 
58  ) 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58  ; 

8 

3902 

3960 

4018 

4076 

4134 

4192 

4250 

4308 

4366 

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58  ) 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4330 

4888 

4945 

5003 

58  J 

750 

875061 

875119 

875177 

875235 

875293 

875351 

375409 

875466 

875524 

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58  ( 

1 

5640 

5698 

5756 

5813 

5871 

5929 

5987 

6045 

6102 

6160 

58 

2 

6218 

6276 

6333 

6391 

6449 

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6564 

6622 

6680 

6737 

58  ' 

58  ) 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

4 

7371 

7429 

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7544 

7602 

7659 

7717 

7774 

7832 

7889 

58  i 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

57  < 

6 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57/ 

7 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

57 

8 

9609 

9726 

9784 

9841 

9898 

9956 

380013 

880070 

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57 

9  880-242 

880299  880356  880413  88047]  880528 

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0(599 1  0756 

57  ! 

No.  |     0      |     1      |     3     |     3 


4      | 


I     «     I     7     |     8 


9      |  Diff. 


O  F  X  V  M  B  E 


131 


3  |  4 


|  9  |  D.tr.- 


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1 
2 

3 


880814 
J385 
1955 
2525 
3093 
3661 
4229 
4795 
5361 
5926 

886491 
7054 
7617 
8179 
8741 
9302 
9862 

890421 
0980 
1537 


2651 
3207 
3762 
4316 
4870 
5423 
5975 
6526 
7077 

897627 
8176 
8725 
9273 
9821 

900367 
0913 
1458 
2003 
2547 

903090 
3633 
4171 
4716 
5256 
5796 
633.1 
6874 
7411 
7949 

908485 
9021 
9556 

910091 
0624 
1158 
1690 
2222 
2753 
3284 


880871 

880J28 

1442 

1499 

2012 

2069 

2581 

2638 

3150 

3207 

3718 

3775 

4285 

4342 

4852 

4909 

5418 

5474 

5983 

6039 

886547 

886604 

7111 

7167 

7674 

7730 

8236 

8292 

8797 

8853 

9358 

9414 

9918 

9974 

890477 

890533 

1035 

1091 

1593 

1649 

892150 

892206 

2707 

2762 

3262 

3318 

3817 

3873 

4371 

4427 

4025 

4980 

5478 

5533 

6030 

6085 

6581 

6(536 

7132 

7187 

897682 

897737 

8231 

8286 

8780 

8835 

9328 

9383 

9875   9930 

900422  900476 

0968 

1022 

1513 

1567 

2057 

2112 

2601 

2655 

903144 

903199 

3687 

3741 

4229 

4283 

4770 

4824 

5310 

5364 

5850 

5904 

6389 

6443 

6927 

6981 

7465 

7519 

8002 

8056 

908539 

908592 

9074 

9128 

9610 

9663 

910144 

910197 

0678 

0731 

1211 

1264 

1743 

1797 

2275 

2328 

2806 

2859 

3337 

3390 

880985 
1556 
2126 
2695 
3264 
3832 
4399 
4965 
5531 
6096 


7223 

7786 
8348 
8009 
9470 


0589 
1147 
1705 

892262 
2818 
3373 
3928 
4482 
5036 
5588 
6140 
6699 
7212 

897792 
8341 
8890 
9437 
9985 

900531 
1077 
1622 
2166 
2710 

903253 
3795 
4337 

4878 
5418 
5958 
6497 
7035 
7573 
8110 

908646 
9181 
9716 

910251 
0784 
1317 
1850 
2381 
2913 
3443 


881042 
1613 
2183 
2752 
3321 
3888 
4455 
5022 
5587 
6152 

886716 
7280 
7842 
8404 
8965 
9526 

890086 
%45 
1203 
1760 

892317 
2873 
3429 
3984 
4538 
5091 
5644 
6195 
6747 
7297 

897847 
8396 
8044 
9102 

900030 
0586 
1131 
1676 
22-21 
2764 

903307 
3849 
4391 
4932 
5472 
6012 
6551 
7089 
7626 
8163 

908699 
9235 
9770 

910304 
0838 
1371 
1903 
2435 
2966 
3496 


881099 
1670 
2240 
2809 
3377 
3945 
4512 
5078 
5644 
6209 

886773 
7336 
7898 
8460 
9021 
9582 

890141 
0700 
1259 
18K3 

892373 
2929 
3484 
4039 
4593 
5146 
5699 
6251 
6802 
7352 

897902 
8451 
8999 
9547 

900004 
0040 
1186 
1731 
2275 
2818 

903361 
3904 
4445 
4986 
5526 
6066 
6604 
7143 
7680 
8217 

908753 
9289 
9823 

910358 
0891 
1424 
1956 
2488 
3019 
3549 


881156 
1727 
2297 
2866 
3434 
4002 
4569 
5135 
5700 
6265 

886829 
7302 
7955 
8516 
9077 
9638 

890197 
0756 
1314 
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892429 
2985 
3540 
4094 
4648 
5201 
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6306 
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8506 
9054 
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900149 
0695 
1240 
1785 
2329 
2873 

903416 
3958 
4499 
5040 
5580 
6119 
6658 
7196 
7734 
8270 

908807 
9342 
9877 

910411 
0944 
1177 
2009 
2541 
3072 
3602 


881213 
1784 
2354 
2923 
3491 
4059 
4625 
5192 
5757 
6321 


7449 
8011 
8573 
9134 
9094 
890253 
0812 
1370 
1928 

892484 
3040 
3595 
4150 
470-1 
5257 
5809 
6361 
6912 
7462 

898012 
8561 
9109 
9656 

900203 
0749 
1295 
1840 
2384 
2927 

903470 
4012 
4553 
5094 
5634 
6173 
6712 
7250 
7787 
8324 

908860 
9396 
9930 

910464 
0998 
1530 
2063 
2594 
3125 
3655 


881271 
1841 
2411 
2980 
3548 
4115 
4682 
5248 
5813 
6378 

886942 
7505 
8067 
8629 
9190 
9750 

890309 
0868 
1426 
1983 

892540 
3096 
3651 
4205 
4759 
5312 
5861 
6416 
6967 
7517 

898067 
8615 
9164 
9711 

900258 
0804 
1349 
1894 
2438 
2981 

903524 
4066 
4607 
5148 
5688 
6227 
6766 
7304 
7841 
8378 

908914 
9449 
9984 

910518 
1051 
1584 
2116 
2647 
3178 
3708 


2468 
3037 
3605 
4172 
4739 
5305 
5870 
6434 


7561 
8123 
8685 
9246 
9806 
890365 
0921 
1482 
2039 

892595 
3151 
3706 
4261 
4814 
5367 
5929 
6471 
7022 
7572 

898122 
8670 
9218 
9760 

900312 
0859 
1404 
1948 
2492 
3036 

903578 
4120 
4661 
5202 
5742 
6281 
6820 
7358 
7895 
8431 

908967 
9503 

910037 
0571 
1104 
1637 
2169 
2700 
3231 
3761 


57 
57 

57 
57 

57 
57 

57 

57  ) 
57 ' 

56 

SB 
56 
56 
56 
56 
56 

56 

56 
56 

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56< 
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55  ( 
55  ( 
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55  ( 
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55  ) 


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132 


LOGARITHMS 


I     0     I     1     I    *    i    3    1    4    J    5    J    6    f   T    t    8    <    9    \ 


Diff) 

53 
53  { 
53 
53  I 
53  ( 
53  ' 
52  ( 


820 
1 

o 

3 
4 
5 


830 
1 
2 
3 
4 
5 


840 
1 
2 
3 
4 
5 
C 
7 


850 
1 
2 
3 
4 
5 
6 


>880 


913814 

913867 

913920 

913973 

914026 

914079 

914132 

914184 

914237 

9142PC 

4343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

4872 

4925 

4977 

5030 

5083 

5136 

5189 

5241 

5294 

5347 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5769 

5822 

5875 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

G980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

7500 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

919078 

919130 

919183 

919235 

919287 

919340 

919392 

919444 

919496 

919549 

9601 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

920019 

920071 

920123 

920176 

920228 

920280 

920332 

920384 

920436 

920489 

0541 

0593 

0045 

0697 

0749 

0801 

0833 

0906 

0958 

1010 

1062 

1114 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

3244 

3296 

3348 

3390 

3451 

3503 

3555 

3607 

3658 

3710 

3762 

3814 

3805 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

924279 

924331 

924383 

924434 

924486 

924538 

924589 

924641 

924693 

924744 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

5312 

5364 

5415 

5467 

5518 

5570   5621 

5673 

5725 

5776 

5828 

5879 

5931 

5982 

6034 

6085   6137 

6188 

6240 

6291 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

6857 

6908 

6959 

7011 

7002 

7114 

7105 

7216 

7268 

7319 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

929419 

929470 

929521 

929572 

929623 

929674 

929725 

929776 

929827 

929879 

9930 

9981 

930032 

930083 

930] 34  930185 

930236 

930287 

930338 

930389 

930440 

930491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

2474 

2524 

2575 

2026 

2677 

2727 

2778 

2829 

2879 

2930 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

4347 

4397 

4448 

934498 

934549 

934599 

934650 

934700 

934751 

934801 

934852 

934902 

934953 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5860 

5910 

5960 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7408 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

9020 

9070 

9120 

8170 

9220 

9270 

9320 

9369 

9419 

9469 

932519 

939569 

939619 

939669 

939719 

939769 

939819 

939869 

939918 

939968 

940018 

940068 

940118 

940168 

940218  940267 

940317 

940307 

940417 

940467 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1S59 

1909 

1958 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2435 

250-1 

2354 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

No.  | 


OP    NUMBERS. 


133 


JnoT  0|1|»|3|4|5|6|T|8|  9|dS] 

)880 

944483  944532 

944581 

944631 

944680 

944729  944779 

944828 

944877 

944927 

49  < 

$  1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

49  ( 

)   2 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

49  ( 

)  3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

49  ( 

S      4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

S   5 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

S   6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49  ( 

<      7 

7924 

7973 

8022 

8070 

8119 

81 68 

8217 

8266 

8315 

8364 

49  I 

^   8 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

49  ( 

(   9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

49  ( 

)  890 

949390 

949439 

949488 

949536 

949585 

949634 

949683 

949731 

949780 

949829 

49  ) 

)   1 

9878 

9926 

9975 

950024 

950073 

950121 

950170 

950219 

950267 

950316 

49  ) 

)   2 

950365 

950414 

950462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

49  ) 

)   3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

49  > 

)   4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

49  > 

/   5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

22(H) 

48  > 

S   6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

43  > 

)   7 

2792 

2841 

2889 

2938 

2980 

3034 

3083 

3131 

3180 

3228 

48  > 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

48  ) 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

48  S 

900 

954243 

954291 

954339 

954387 

954435 

954484 

954532 

954580 

954628 

954677 

48  \ 

?   1 

4725 

4773 

4821 

4869 

4913 

4966 

5014 

5062 

5J10 

5158 

48  ( 

<   2 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48  ( 

1   3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

48  ( 

1   4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48  ( 

I      5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48  < 

/   6 

7128 

7176 

7224 

7272 

7320 

7966 

7416 

7464 

7512 

7559 

48  ( 

?   7 

WOT 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48  I 

)   8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48  ( 

)   9 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

48  J 

<910 

959041 

959089 

959137 

959185 

959232 

959280 

959328 

959375 

959423 

959471 

48  ) 
48  , 

1 

9518 

9566 

9614 

9661 

9109 

9757 

9804 

9852 

9900 

9947 

2 

9995 

960042 

960090 

960138 

960185 

960233 

960281 

960328 

960376 

960423 

48  ) 

S   3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48  ) 

S   4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

48  ) 

S   5 

1421 

1469 

1516 

1563 

1011 

1658 

1706 

1753 

1801 

1848 

47  ) 

C   6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47  ) 

7 

2369 

2417 

2464 

2511 

2559 

2G06 

2653 

2701 

2748 

2795 

47  > 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47  S 

9 

3316 

3303 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

47  S 

920 

963788 

963835 

963882 

963929 

963977 

964024 

964071  964118 

964165 

964212 

47  / 

1 

4860 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4(584 

47  / 

I      2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

47  / 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

47  ; 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

47  ) 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

47  ) 

)   6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

47 
47  ) 

)   7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

;  8 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

47 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

47  \ 

{930 

968483 

968530 

968576 

968623 

968670 

968716 

968763 

968810 

968856 

968903 

47  ( 

{   1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

47  < 

I      2 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47  < 

(   3 

9882 

9928 

9975 

970021 

970068 

970114 

970161 

970207 

970254 

970300 

47  < 

(   4 

970347 

970393 

970440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46  ' 

)   5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

46^ 

{   6 
I      7 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

46  < 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

46  < 

5   8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

46  ( 

9 

2666 

2712 

2758 

2804 

2851 

28^7 

2943 

2989 

3035 

3082 

46  j 

,£0-^L-3~LJL~LJLJ  ^^L  JL  XJLX~S  JLs 


134 


LOGARITHMS,     ETC. 


■;m 


973128 
3590 
4051 
4512 
4972 
5432 
5891 
6350 
6808 
7266 

977724 
8181 
8637 
9093 
9548 

980003 
0458 
0912 
1366 
1819 

982271 
2723 
3175 
3026 
4077 
4527 
4977 
5426 
5875 
6324 

970  986772 


7219 
7666 
8113 
8559 
9005 
9450 
9895 
990339 
0783 

991220 
1669 
2111 
2554 
2995 
3436 
3877 
4317 
4757 
519(5 

995035 
6074 
6512 
6949 
7386 
7823 
8259 
8695 
9131 
9565 


973174 

973220 

3636 

3682 

4097 

4143 

4558 

4604 

5018 

5064 

5478 

5524 

5937 

5983 

6396 

6442 

6854 

6900 

7312 

7358 

977769 

977815 

8226 

8272 

8683 

8728 

9138 

9184 

9594 

9639 

980049 

980094 

0503 

0549 

0957 

1003 

1411 

1456 

1864 

1909 

982316 

982362 

2769 

2814 

3220 

3265 

3671 

3716 

4122 

4167 

4572 

4617 

5022 

5067 

5471 

5516 

5920 

5965 

6369 

6413 

986817 

986861 

7264 

7309 

7711 

7756 

8157 

8202 

8604 

8648 

9049 

9094 

9494 

9539 

9939 

9983 

990383 

990428 

0827 

0871 

991270 

991315 

3713 

1758 

2156 

2200 

2598 

2642 

3039 

3083 

3480 

3524 

3921 

3965 

4361 

4405 

4801 

4^45 

5240 

5284 

995679 

995723 

6117 

6161 

6555 

6599 

6993 

7037 

7430 

7474 

7867 

7910 

8303 

8347 

8739 

8782 

9174 

9218 

9609 

9652 

973266 
3728 
4189 
4650 
5110 
5570 
6029 
6488 
6946 
7403 

977861 
8317 
8774 
9230 
9685 

980140 
0594 
1048 

1:01 

1954 

982407 
2859 
3310 
3762 
4212 
4662 
5112 
5561 
6010 
6458 


7353 

7800 
8247 
8693 
9138 
9583 
990028 
0472 
0916 

991359 
1802 
2244 
2686 
3127 
3568 
4009 
4449 
4889 
5328 

995767 
6205 
6643 
7080 
7517 
7954 
8390 
8826 
9261 
9696 


73313 
3774 
4235 
4696 
5156 
5616 
6075 
6533 
6992 
7449 

977906 
8363 
8819 
9275 
9730 

980185 
0640 
1093 
1547 
2000 

982452 
2904 
3356 
3807 
4257 
4707 
5157 
5606 
6055 
6503 

986951 
7398 
7845 
8291 
8737 
9183 
9628 

990072 
0510 
0960 

991403 
1846 
2288 
2730 
3172 
3613 
4053 
4493 
4933 
5372 

99581] 
6249 
6687 
7124 
7561 
7998 
8434 
8869 
9305 
9739 


973359 
3820 
4281 
4742 
5202 
5662 
6121 
6579 
7037 
7495 

977952 
8409 
8865 
9321 
9776 

980231 
0685 
1139 
1592 
2045 

982497 
2949 
3401 
3852 
4302 
4752 
5202 
5651 
6100 
6548 

986996 
7443 
7890 
8336 
8782 
9227 
9672 

990117 
0561 
1004 

991448 
1890 
2333 
2774 
3216 
3657 
4097 
4537 
4977 
5416 

995854 
6293 
6731 
7168 
7605 
8041 
8477 
8913 
934H 
9783 


734051973451 


3866 
4327 
4788 
5248 
5707 
6167 
6625 
7083 
7541 

977998 
8454 
8911 
9366 
9821 

980276 
0730 
1184 
1637 
2090 

982543 
2994 
3446 
3897 
4347 
4797 
5247 
5696 
6144 
6593 

987040 
7488 
7934 
8381 
8826 
9272 
9717 

990161 
0605 
1049 

991492 
1935 
2377 
2819 
3260 
3701 
4141 
4581 
5021 
5460 

995898 
6337 
6774 
7212 
7648 
8085 
8521 
8956 
9TO2 
9826 


3913 
4374 
4834 
5294 
5753 
6212 
6671 
7129 
7586 

978043 
8500 
8956 
9412 
9867 

980322 
0776 
1229 
1683 
2135 

982588 
3040 
3491 
3942 
4392 
4842 
5292 
5741 
6189 
6637 

987085 
7532 
7979 
8425 
8871 
9316 
9761 

990206 
0650 
1093 

991536 
1979 
2421 
2863 
3304 
3745 
4185 
4625 
5065 
5504 

995942 
6380 
6818 
7255 
7692 
8129 
8564 
9000 
9435 
9870 


973497  973543 

3959 

4005 

4420 

4466 

4880 

4926 

5340 

5386 

5799 

5845 

6258 

6304 

6717 

6763 

7175 

7220 

7632 

7678 

978089 

978135 

8546 

8591 

9002 

9047 

9457 

9503 

9912 

9958 

980367 

980412 

0821 

0867 

1275 

1320 

1728 

1773 

2181 

2226 

982633 

982678 

3085 

3130 

3536 

3581 

3987 

4032 

4437 

4482 

4887 

4932 

5337 

5382 

5786 

5830 

6234 

6279 

6682 

6727 

987130 

987175 

7577 

7622 

8024 

8068 

8470 

8514 

8916 

8960 

9361 

9405 

9806 

9850 

990250 

990294 

0694 

073« 

1137 

1182 

991580 

991625 

2023 

2067 

2465 

2509 

2907 

2951 

3348 

3392 

3789 

3833 

4229 

4273 

4669 

4713 

5108 

5152 

5547 

5591 

995986 

996030 

6424 

6468 

6862 

6906 

7299 

7343 

7736 

7779 

8172 

8216 

8608 

8652 

9043 

9087 

9479 

9522 

9913 

91)57 

l^-LjLXJLJ^Ji^JL  ^  •'  JLJ.- 


LOGARITHMIC 

SINES  AND   TANGENTS, 


FOR    EVERY 


DEGREE  AND  MINUTE 


THE    QUADRANT. 


N.  B.  Thk  minutes  in  the  left-hand  column  of  each  pagfc; 
increasing  downwards,  belong  to  the  degrees  at  the  top ;  and  those 
increasing  upwards,  in  the  right-hand  column,  belong  to  the  degree 
below. 


13t 

1     (0  Degree.) 

A  TABLE  OJ 

*     LOGARITHMIC 

nr 

|    Sine 

1   D. 

Cosine 

1  D. 

|   Tan?. 

1   D. 

|   Cotansr. 

'H 

(  ° 

0-000000 

10-000000 

o-oooooo 

1 

Infinite. 

60 ; 

/  i 

6463726 

501717 

000000 

00 

0-463726 

1  501717 

13-536274 

59? 

(  2 

764756 

293485 

000000 

00 

764756 

293483 

235244 

58 ; 

/  3 

940847 

208231 

000000 

00 

940847 

208231 

059153 

57  ) 

I    4 

7-065786 

161517 

000000 

00 

7-065786 

161517 

12-934214 

56  > 

(  5 

162696 

131968 

000000 

00 

162696 

131969 

837304 

55  ) 

(  6 

241877 

111575 

9-999999 

01 

241878 

111578 

758122 

54  ) 

/  V 

308824 

96653 

999999 

01 

308825 

99653 

691175 

53  ) 

t    8 

366816 

85254 

999999 

01 

366817 

85254 

633183 

52  ) 

<    9 

417968 

76263 

999999 

01 

417970 

76263 

582030 

51  ) 

463725 

68988 

999998 

01 

463727 

68988 

536273 

50  / 
49  ( 

\  ]1 

7-505118 

62981 

9-999998 

01 

7-505120 

62981 

12-494880 

( 12 

542906 

57936 

999997 

01 

542909 

57933 

457091 

43  ( 

S  13 

577668 

53641 

999997 

01 

577672 

53642 

422328 

47  ( 

\  u 

609853 

49938 

999996 

01 

609857 

49939 

390143 

46  ? 

I  15 

639816 

46714 

999996 

01 

639820 

46715 

360180 

45  ( 

M6 

667845 

43881 

999995 

01 

667849 

43882 

332151 

44  ( 

<! 17 

694173 

41372 

999995 

01 

694179 

41373 

305821 

43  ( 

<  18 

718997 

39135 

999994 

01 

719003 

39136 

280997 

42  / 

<  19 

742477 

37127 

999993 

01 

742484 

37128 

257516 

41  ? 

jao 

764754 

35315 

999993 

01 

764761 

35136 

235239 

40  ( 
39  ) 

<21 

7-785943 

33672 

9-999992 

01 

7-785951 

33C73 

12-214049 

(  22 

806146 

32175 

999991 

01 

806155 

32176 

193845 

38  < 

(  23 

825451 

30805 

999990 

01 

825460 

30806 

174540 

37  ( 

<24 

843934 

29547 

999989 

02 

843944 

29549 

156056 

36  ( 

(25 

861662 

28388 

999988 

02 

801674 

28390 

138326 

35  ( 

<26 

878695 

27317 

999988 

02 

878708 

27318 

121292 

34  ( 

(27 

895085 

26323 

999987 

02 

895099 

20325 

104901 

33  ( 

(28 

910879 

25399 

999986 

02 

910894 

25401 

089106 

32  ( 

(  29 

926119 

24538 

999985 

02 

926134 

24540 

073866 

31  ( 

(30 

940842 

23733 

999983 

02 

940858 

23735 

059142 

30  j 

)31 

7-955082 

22980 

9-999982 

02 

7-955100 

22981 

12044900 

29  ) 

)32 

968870 

22273 

999981 

02 

968889 

22275 

031111 

28  ( 

>33 

982233 

21608 

999980 

02 

982253 

21610 

017747 

27 

I34 

995198 

20981 

999979 

02 

995219 

20983 

004781 

26  ) 

)35 

8-007787 

20390 

999977 

02 

8-007809 

20392 

11-992191 

25  S 

S36 

020021 

19831 

999976 

02 

020045 

19833 

979955 

24  ) 

)37 

031919 

19302 

999975 

02 

031945 

19305 

908055 

23  ( 

38 

043501 

18801 

399973 

02 

043527 

18803 

956473 

22  ( 

)39 

054781 

18325 

999972 

02 

054809 

18327 

945191 

21  ( 

Wo 

065776 

17872 

999971 

02 

065806 

17874 

934194 

20  ( 

42 

8-076500 

17441 

9-999969 

02 

8-076531 

17444 

11-923469 

19  ) 

0869C5 

17031 

999968 

02 

080997 

17034 

913003 

18  ) 

}43 

097183 

16639 

999966 

02 

097217 

16642 

902783 

17  ) 

>44 

107167 

16265 

999964 

03 

107202 

16268 

892797 

15  ; 

)45 

116926 

15908 

999963 

03 

116963 

15910 

883037 

>46 

126471 

15566 

999961 

03 

126510 

15568 

873490 

14  > 

)47 

135810 

15238 

999959 

03 

135851 

15241 

864149 

13  ) 

)48 

144953 

14924 

999958 

f,3 

144996 

14927 

855004 

12) 

)49 

153907 

14622 

999956 

03 

153952 

14627 

846048 

Jl  ) 

)50 

162681 

14333 

999954 

03 

162727 

14336 

837273 

10  j 

(51 

8171280 

14054 

9-999952 

03 

8-171328 

14057 

11-828672 

9  ( 

(52 

179713 

13786 

999950 

03 

179763 

13790 

820237 

8  ( 

(53 

187985 

13529 

999948 

03 

188036 

13532 

811964 

7  ) 

(54 

196102 

13280 

999946 

03 

196156 

13284 

803844 

G  I 

(55 

204070 

13041 

999944 

03 

204126 

13044 

795874 

5  ) 

(56 

211895 

12810 

999942 

04 

211953 

12814 

788047 

4  ) 

(57 

219581 

12587 

999940 

04 

219641 

12590 

780359 

3  > 

58 

50 

'  60 

227134 

12372 

999938 

04 

227195 

12376 

772805 

2  ) 

234557 

12164 

999936 

04 

234621 

12168 

765379 

1  ) 

241855 

11963 

999934 

04 

241921 

11967 

758079 

0  > 

L. 

Cosine 

^^J 

^Ji^J 

~J 

Cotang.   | 

^^^^i 

->*>£3L*J. 

89  1 

.Jegre 

Mi 

sines    and    tangents.     (1  Degree,) 


137 


I  m. 

Sine 

~  D. 

Cosine 

D. 

Tang, 

D.    | 

Cotang. 

1 

i  o 

8241855 

11963 

9999934 

04 

8241921 

11967 

11-758079 

60  ( 

1 

249033 

11768 

999932 

04 

249102 

11772 

750898 

59  ( 

2 

256094 

11580 

999929 

04 

256165 

11584 

743835 

58? 

3 

263042 

11398 

999927 

04 

263115 

11402 

736885 

57  { 

4 

269881 

11221 

999925 

04 

269956 

11225 

730044 

56  ( 

)    5 

27(3614 

11050 

999922 

04 

276691 

11054 

723309 

55  ( 

)    6 

283243 

10883 

999920 

04 

283323 

10887 

716677 

54  I 

)    7 

289773 

10721 

999918 

04 

289856 

10726 

710144 

53  { 
52  ( 
51  ( 

<  8 

296207 

10565 

999915 

04 

296292 

10570 

703708 

)    9 

302546 

10413 

999913 

04 

302634 

10418 

697366 

)  10 

308794 

10266 

999910 

04 

308884 

10270 

691116 

50  I 

)   H 

8-314954 

10122 

9-999907 

04 

8-315046 

10126 

11-684954 

49  S 

)  12 

321027 

9982 

999905 

04 

321122 

9987 

678878 

48  ) 

)  13 

327016 

9847 

999902 

04 

327114 

9851 

672886 

47  S 

)  14 

332924 

9714 

999899 

05 

333025 

9719 

666975 

46  S 

)  15 

338753 

9586 

999897 

05 

338856 

9590 

601144 

45  ) 

>  16 

344504 

9460 

999894 

05 

344610 

9405 

655390 

44  ) 

)  1? 

350181 

9338 

999891 

05 

350289 

9343 

649T1 1 

43  ; 

)  I8 

355783 

9219 

999888 

05 

355895 

9224 

644105 

42  ) 

)  1(J 

361315 

9103 

999885 

05 

361430 

9108 

638570 

41  ) 

(20 

360777 

8990 

999882  . 

05 

366895 

8995 

633105 

40  S 

?21 

8-372171 

8880' 

9-999879 

05 

8-372292 

8885 

11-627708 

39  I 

22 

377499 

8772 

999876 

05 

377622 

8777 

622378 

38  > 

)  23 

382762 ' 

8667 

999873 

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382889 

8672 

617111 

37  ) 

)  24 

387962 

8564 

999870 

05 

388092 

8570 

611908 

36  > 

)  2-5 

393101 

8464 

999867 

05 

393234 

8470 

606766 

35  ) 

)  2(i 

368179 

8366 

999864 

05 

398315 

8371 

601685 

34  > 

)  27 

403199 

8271 

999861 

05 

403338 

8276 

590602 

33  ) 

)  28 

408161 

8177 

999858 

05 

408304 

8182 

591696 

32  ) 

>  29 

413068 

8086 

999854 

05 

413213 

8091 

586787 

31  > 

S   30 

417919 

7996 

999851 

06 

418068 

8002 

581932 

30  ) 

(  31 

8422717 

7909 

9-999848 

06 

8-422869 

7914 

11-577131 

29  > 

)  32 

427462 

'7823 

999844 

06 

427618 

7830 

572382 

28  ( 

)  33 

4:12 1 56 

7740 

999841 

06 

432315 

7745 

567685 

27  ( 

)  34 

436800 

7657 

999838 

06 

436962 

7663 

563038 

26  ( 

)  35 

441394 

7577 

999834 

06 

441560 

7583 

558440 

25  ( 

)  36 

44.i941 

7499 

999831 

06 

446110 

7505 

553890 

24  ( 

>  37 

4504-10 

7422 

999827 

06 

450613 

7428 

549387 

23  ( 

)  38 

45-1893 

7346 

999823 

06 

455070 

7352 

544930 

22  ( 

)  39 

459301 

7273 

999820 

06 

459481 

7279 

540519 

21  ( 

j  40 

463665 

7200 

999816 

06 

463849 

7206 

536151 

20  ( 

(  41 

8-467985 

7129 

9-999812 

06 

8-468172 

7135 

11531828 

19  > 

(  42 

472263 

7060 

999809 

06 

472454 

7066 

527546 

18  ) 

(  43 

476498 

6991 

999805 

06 

476693 

6998 

523307 

17  S 

<  44 

480693 

6924 

999801 

06 

480892 

6931 

519108 

16  ) 

(  45 

484848 

6859 

999797 

07 

485050 

6865 

514950 

15  ) 

?46 

488963 

6794 

999793 

07 

489170 

6801 

510830 

14  ) 

/47 

493040 

6731 

999790 

07 

493250 

6738 

506750 

13  S 

/48 

497078 

6669 

999786 

07 

497293 

6676 

51,2707 

12  ) 

I  49 

501080 

6608 

999782 

07 

501298 

6615 

498702 

11  ) 

)  50 

505045 

6548 

999778 

07 

505267 

6555 

494733 

10  \ 

I  51 

8-5(i8974 

6489 

9-999774 

07 

8-509200 

6496 

11-490800 

9  ) 

(  52 

512867 

6431 

999769 

07 

513098 

6439 

480902 

8/ 

I  53 

516726 

6375 

999765 

07 

516961 

6382 

483039 

7  ) 

(  54 

520551 

6319 

999761 

07 

520790 

6326 

479210 

6? 

/  55 

524343 

6264 

999757 

07 

524586 

6272 

475414 

5  ) 

(  56 

528102 

6211 

989753 

07 

528349 

6218 

471651 

4  ) 

(  57 

531828 

6158 

999748 

07 

532080 

6165 

467920 

3  ) 

(58 

535523 

6106 

999744 

07 

535779 

6113 

464221 

2  ) 

(59 

539186 

6055 

999740 

07 

539447 

6062 

460553 

1  ) 

(60 

542819 

6004 

999735 

07 

543084 

6012 

456916 

0  J 

Cosine   | 

^~^J. 

Sine 

~-^J 

Cotang. 

L^~^ 

^^Tang^^ 

J^i 

Degrees. 


138 


(2  Degrees.)     a    table    OP    logarithmic 


r»«r 

Sine 

D. 

Cosine 

D. 

Tan*. 

1   D. 

Cotanj. 

H 

1 

8-542819 

6004 

9999735 

07 

8  543084 

6012 

11-456916 

60? 

546422 

5955 

999731 

07 

546691 

5962 

453309 

59  I 

549995 

5906 

999726 

07 

550268 

5914 

449732 

58 

\  3 

553539 

5858 

999722 

08 

553817 

5866 

446183 

57  < 

)    4 

557054 

5811 

999717 

08 

557336 

5819 

442664 

56  ( 

S  5 

560540 

5765 

999713 

08 

560828 

5773 

439172 

55  ( 

54  I 

)  6 

563999 

5719 

999708 

08 

564291 

5727 

435709 

)    7 

567431 

5674 

990704 

08 

567727 

5682 

432273 

S3( 
52  } 

)  8 

570836 

5630 

999699 

08 

571137 

5638 

428863 

}  9 

574214 

5587 

999694 

08 

574520 

5595 

425480 

51  | 

<  10 

577566 

5544 

999689 

08 

577877 

5552 

422123 

50  l 

!  n 

8-580892 

5502 

9-999685 

08 

8581208 

5510 

11-418792 

49  j 

)  12 

584193 

5460 

999680 

08 

584514 

5468 

415486 

48  < 

)  13 

587469 

5419 

999675 

08 

587795 

5427 

412205 

47  < 

)  W 

590721 

5379 

999670 

08 

591051 

5387 

408949 

46  < 

)  15 

593948 

5339 

999665 

08 

594283 

5347 

405717 

45  < 

)  16 

597152 

5300 

999660 

08 

597492 

5308 

402508 

44  < 

)  17 

600332 

5261 

999655 

08 

600677 

5270 

399323 

43  S 

)  18 

603489 

5223 

999650 

08 

603839 

5232 

396161 

42  < 

)  1!> 

606623 

5186 

999645 

09 

606978 

5194 

393022 

41  i 

j  20 

609734 

5149 

999640 

09 

610094 

5158 

389906 

40  s 

}  21 

8-612823 

5112 

9-999635 

09 

8-613189 

5121 

11-386811 

39  ) 

)  22 

615891 

5076 

999629 

09 

616262 

5085 

383738 

38  ) 

>  23 

618937 

5041 

999624 

09 

619313 

5050 

380687 

37  ) 

)  24 

621962 

5006 

999619 

09 

622343 

5015 

377657 

36  ) 

>25 

624965 

4972 

999614 

09 

625352 

4981 

374648 

35  ) 

)  96 

627948 

4938 

999608 

09 

628340 

4947 

371660 

34  ) 

)  27 

630911 

4904 

999603 

09 

631308 

4913 

368692 

33  ) 

)  28 

633854 

4871 

999597 

09 

634256 

4880 

365744 

32  ) 

)  29 

636776 

4839 

999592 

09 

637184 

4848 

362816 

31  ) 

j  30 

639680 

4806 

999586 

09 

640093 

4816 

359907 

30  j 

>31 

8642563 

4/75 

9-999581 

09 

8642982 

4784 

11-357018 

29  ) 

(  32 

645428 

4743 

999575 

09 

645853 

4753 

354147 

28  ( 

(  33 

648274 

4712 

999570 

09 

648704 

4722 

351296 

27  ? 

(  34 

651102 

4682 

999564 

09 

651537 

4C91 

348413 

26  / 

(   35 

653911 

4652 

999558 

10 

654352 

4661 

345648 

25  ( 

I  36 

656702 

4622 

999553 

10 

657149 

4631 

342851 

24  ) 

)  37 

659475 

4592 

999547 

10 

659928 

4602 

340072 

23  ? 

)  38 

662230 

4563 

999541 

10 

662689 

4573 

337311 

22  ( 

I   39 

664968 

4535 

999535 

10 

665433 

4544 

334567 

21  ? 

)  40 

667689 

4506 

999529 

10 

668160 

4526 

331840 

20  / 

i  41 

8-670393 

4479 

9-999524 

10 

8-670870 

4488 

11-329130 

19  ( 

<  42 

673080 

4451 

999518 

10 

673.563 

4461 

326437 

18  < 

43 
(  44 
I  45 

675751 

4424 

999512 

10 

676239 

4434 

323761 

8} 

678405 

4397 

999506 

10 

678900 

4417 

321100 

681043 

4370 

999500 

10 

681544 

4380 

318456 

(  46 

683665 

4344 

999493 

10 

684172 

4354 

315828 

S  47 

686272 

4318 

999487 

10 

686784 

4328 

313216 

13 

(  48 

688863 

4292 

999481 

10 

689381 

4303 

310619 

12  f 

(  49 

691438 

4267 

999475 

10 

691963 

4277 

308037 

ii , 

(   50 

693998 

4242 

999469 

10 

694529 

4252 

305471 

10 1 

9 

i51 

8-696543 

4217 

9-999463 

11 

8-697081 

4228 

11-302919 

52 
53 

699073 

4192 

999456 

11 

699617 

4203 

300383 

8( 

701589 

4168 

999450 

11 

702139 

4179 

297861 

7  . 

$54 

704090 

4144 

999443 

11 

704646 

4155 

295354 

! 

)  55 

706577 

4121 

999437 

11 

707140 

4132 

292800 

)   56 

709049 

4097 

999431 

11 

709618 

4108 

290382 

4  ) 

)   57 

711507 

4074 

999424 

11 

712083 

4085 

287917 

3  ) 

<  58 

713952 

4051 

999418 

11 

714534 

4062 

285465 

2  ) 

<  59 

716383 

4029 

999411 

11 

716972 

4040 

283028 

1  ) 

J  60 

718800 

4006 

999404 

11 

719396 

4017 

280604 

0  j 

LwJ 

Cosine   | 

^ | 

^-2^!~4 

Cotang. 

„ 1 

^J^Ek^. 

J^J 

87  Decrees. 


SINES    AND    TANGENTS.    (3  Degrees.) 


139 


r»L 

Sine 

D. 

|   Cosine 

D. 

Tang. 

D. 

Cotang. 

1     ) 

(  ° 

8-718800 

4006 

9-999404 

11 

8-719396 

4017 

11-280604 

i  60  I 

?  i 

721204 

3984 

999398 

11 

721806 

3995 

278194 

59  } 

i  2 

723595 

3962 

999391 

11 

724204 

3974 

275796 

58  ( 

3 

725972 

3941 

999384 

11 

726588 

3952 

273412 

57  ' 

4 

728337 

3919 

999378 

11 

728959 

3930 

271041 

56  ) 

i    5 

730688 

3898 

999371 

11 

731317 

3909 

268683 

55  ) 

)  6 

733027 

3877 

9993(34 

12 

733663 

3889 

266337 

54  ) 

?  7 

735354 

3857 

999357 

12 

735996 

3868 

264004 

53  ) 

)  8 

737667 

3836 

999350 

12 

738317 

3848 

261683 

52  ) 

)  9 

739969 

3816 

999343 

12 

740626 

3827 

259374 

51  ) 

\   10 

742259 

3796 

999336 

12 

742922 

3807 

257078 

50  J 

<  U 

8744536 

3776 

9-999329 

12 

8-745207 

.  3787 

11.254793 

49  > 

(  12 

746802 

3756 

999322 

12 

747479 

3768 

252521 

48  > 

(  I3 

749055 

3737 

999315 

12 

749740 

3749 

250260 

47  > 

(  I4 

751297 

3717 

999308 

12 

751989 

3729 

248011 

46  ) 
45  ) 

)   15 

753528 

3698 

999301 

12 

754227 

3710 

245773 

)   16 

755747 

3679 

999294 

12 

756453 

3692 

243547 

44 

(  I7 

757955 

3661 

999286 

12 

758668 

3673 

241332 

43 

)  18 

760151 

3642 

999279 

12 

760872 

3655 

239128 

42  \ 

)  19 

762337 

3624 

999272 

12 

763065 

3636 

236935 

41  ) 
40  ', 

)  20 

764511 

3606 

999265 

12 

765246 

3618 

234754 

\   21 

8-766675 

3588 

9-999257 

12 

8-767417 

3600 

11232583 

39  ) 

(  22 

768828 

3570 

999250 

13 

769578 

3583 

230422 

38  \ 

(  23 

770970 

3553 

999242 

13 

771727 

3565 

228273 

37  i 

(  24 

773101 

3535 

999235 

13 

773866 

3548 

226134 

36  ^ 

?  25 

775223 

3518 

999227 

13 

775995 

3531 

224005 

35) 

?  26 

777333 

3501 

999220 

13 

778114 

3514 

221886 

34  S 

?  27 

779434 

3484 

999212 

13 

780222 

3497 

219778 

33  S 

?  28 

781524 

3467 

999205 

13 

782320 

3480 

217680 

32  ) 

?  29 

783605 

3451 

999197 

13 

784408 

3464 

215592 

31  ) 

j  30 

785675 

3431 

999189 

13 

786486 

3447 

213514 

30  S 

<  31 

8-787736 

3418 

9-999181 

13 

8-788554 

3431 

11-211446 

29  > 

(  32 

789787 

3402 

999174 

13 

790613 

3414 

209387 

28  ) 

(  33 

791828 

3386 

999166 

13 

792662 

3399 

207338 

27  ) 

(  34 

793859 

3370 

999158 

13 

794701 

3383 

205299 

26  ) 

<  35 

795881 

3354 

999150 

13 

796731 

3368 

203269 

25  } 

(  36 

797894 

3339 

999142 

13 

798752 

3352 

201248 

24  ) 

'  37 

799897 

3323 

999134 

13 

800763 

3337 

199237 

23  ) 

k   38 
(  39 

801892 

3308 

999126 

13 

802765 

3322 

197235 

22  ) 

803876 

3293 

999198 

13 

804758 

3307 

195242 

21  ) 

J  40 

805852 

3278 

999110 

13 

806742 

3292 

193258 

20 

)  41 

8-807819 

3263 

9-999102 

13 

8-808717 

3278 

11191283 

19  } 

S  42 

809777 

3249 

999094 

14 

810683 

3262 

189317 

18  > 

\  43 

811726 

3234 

999086 

14 

812641 

3248 

187359 

17  > 

S  44 

813667 

3219 

999077 

14 

814589 

3233 

185411 

16  ) 

5  45 

815599 

3205 

999069 

14 

816529 

3219 

183471 

15  ) 

\  46 

817522 

3191 

999061 

14 

818461 

3205 

181539 

14  ) 

(  47 

819436 

3177 

999053 

14 

820384 

3191 

179616 

13  > 

<  48 

821343 

3163 

999044 

14 

822298 

3177 

177702 

12  > 

(   49 

823240 

3149 

999036 

14 

824205 

3163 

015195. 

11  ) 

<  50 

825130 

3135 

999027 

14 

826103 

3150 

173897 

10  } 

}  51 

8-827011 

3122 

9-999019 

14 

8-827992 

3136 

11172008 

9  I 

)  52 

828884 

3108 

999010 

14 

829874 

3123 

170126 

8  ( 

)  53 

830749 

3095 

999002 

14 

831748 

3110 

168252 

7 

)  54 

832607 

3082 

998993 

14 

833613 

3096 

166387 

6 

)  55 

834456 

3069 

998984 

14 

835471 

3083 

164529 

5  ( 

)   56 

836297 

3056 

998976 

14 

837321 

3070 

162679 

4  ( 

>  57 

838130 

3043 

998967 

15 

839163 

3057 

160837 

3  I 

S  58 

839956 

3030 

998958 

15 

840998 

3045 

159002 

2  I 

)   59 

841774 

3017 

998950 

15 

842825 

3032 

157175 

1  > 

j  60 

843585 

3000 

998941 

15 

844644 

3019 

155356 

° 

L~. 

Coame 

~>*~*~ 

J    Sine 

.^-^ 

Coiaiig. 

Tang. 

*y 

86  Degrees. 


140 

(4  Degrees.)  a  TABLE  OP  LOGARITHMIC 

(TT 

|    Sme 

D. 

Cosine 

D. 

1   Tang. 

1   D. 

Cotang. 

^^ 

)   o 

8843585 

3005 

9-998941 

15 

8-844644 

3019 

11155356 

60 

;   1 

845387 

2992 

998932 

15 

846455 

3007 

153545 

59 

)  2 

847183 

2980 

998923 

15 

848260 

2995 

151740 

58 

)  3 

848971 

2967 

998914 

15 

850057 

2982 

149943 

57 

?    4 

830751 

2955 

998905 

15 

851846 

2970 

148154 

56 

;   5 

832523 

2943 

998896 

15 

853628 

2958 

146372 

55 

J  G 

854291 

2931 

998887 

15 

855403 

2946 

144597 

54 

)  ~ 

856049 

2919 

998878 

15 

857171 

2935 

142829 

53 

)  8 

837801 

2907 

998869 

15 

858932 

2923 

141068 

52 

)  9 

859546 

2896 

998860 

15 

860686 

2911 

139314 

51 

( 10 

861283 

2884 

998851 

15 

862433 

2900 

137567 

50 

} J1 

8863014 

2873 

9-998841 

15 

8-864173 

2888 

11135827 

49 

)  12 

864738 

2861 

998832 

15 

865906 

2877 

134094 

48 

)  13 

866455 

2850 

998823 

16 

867632 

2866 

132308 

47 

)  14 

868165 

2839 

998813 

16 

869351 

2854 

130649 

46 

(  15 

869868 

2S28 

998804 

16 

871064 

2843 

128936 

45 

)  16 

871565 

2817 

998795 

16 

872770 

2832 

127230 

44 

)  17 

873255 

2806 

998785 

16 

874469 

2821 

125531 

43 

(   18 

874938 

2795 

998776 

10 

8761G2 

2811 

123S33 

42 

)  19 

876615 

2786 

998766 

16 

877849 

2800 

122151 

41 

>  20 

878285 

2773 

998757 

16 

879529 

2789 

120471 

40 

(  21 

8879949 

2763 

9-998747 

16 

8-881202 

2779 

11-118798 

39 

(  22 

881607 

2752 

998738 

16 

882869 

2768 

117131 

38 

<  23 

883238 

2742 

998728 

16 

884530 

2758 

115470 

37 

(  24 

884903 

2731 

998718 

16 

886185 

2747 

113815 

36 

(  23 

886542 

2721 

998708 

16 

887833 

2737 

112167 

35 

(  2(5 

888174 

2711 

998G99 

16 

889476 

2727 

110524 

34 

(  27 

880801 

2700 

998689 

16 

891112 

2717 

108888 

33 

}  28 

891421 

2690 

998G79 

16 

892742 

2707 

107258 

32 

<  29 

893035 

2680 

998669 

17 

894366 

3697 

105634 

31 

I  30 

894643 

2670 

998659 

17 

895984 

2687 

104016 

30 

)  31 

8-896246 

2660 

9-998649 

17 

8-897596 

2677 

11-102404 

29 

S  32 

897842 

2651 

998639 

17 

899203 

2667 

100797 

28 

<  33 

899432 

2641 

998629 

17 

900803 

2658 

099197 

27 

)  34 

901017 

2631 

998619 

17 

902398 

2648 

097602 

26 

(  35 

902596 

2622 

998609 

17 

903987 

2638 

096013 

25 

S  3G 

904169 

2612 

998599 

17 

905570 

2629 

094430 

24 

<  37 

905736 

2603 

998589 

17 

907147 

2620 

092853 

23 

<  38 

907297 

2593 

998578 

17 

908719 

2610 

091281 

22 

<  39 

908853 

2584 

9985G8 

17 

910285 

2601 

089715 

21 

J  40 

910404 

2575 

998558 

17 

911846 

2592 

088154 

20 

)  41 

8-911949 

2566 

9-998548 

17 

8-913401 

2583 

11-086599 

19 

)  42 

913488 

2556 

998537 

17 

914951 

2574 

085049 

18 

)  43 

915022 

2547 

998527 

17 

916495 

2565 

083505 

17 

)  44 

910550 

2538 

998516 

18 

918034 

2556 

081966 

16 

)  43 

918073 

2529 

998506 

18 

919568 

2547 

080432 

15 

>  46 

919591 

2520 

998495 

18 

921096 

2538 

078904 

14 

)  47 

921103 

2512 

998485 

18 

922619 

2530 

077381 

13 

)  48 

922610 

2503 

998474 

18 

924136 

2521 

075864 

12 

$50 

924112 

2494 

998464 

18 

925049 

2512 

074351 

11 

925609 

2486 

998453 

18 

927156 

2503 

072844 

10 

>51 

8-927100 

2477 

9-998442 

18 

8-928658 

2495 

11-071342 

9 

fSQ 

928387 

2409 

998431 

18 

930155 

2486 

069845 

8 

l  S3 

)  54 

930068 

2460 

998421 

18 

931647 

2478 

068353 

7 

931544 

2452 

998410 

18 

933134 

2470 

006866 

6 

/  55 

933015 

2443 

998399 

18 

934616 

2461 

065384 

5 

/56 

934481 

2435 

998388 

18 

936093 

2453 

063907 

4 

.  57 

935942 

2427 

998377 

18 

937565 

2445 

062435 

3 

;  58 

)59 

937398 

2419 

998366 

38 

939032 

2437 

060968 

2 

938850 

2411 

998355 

18 

940494 

2430 

059506 

1 

;  oo 

94(1296 

2403 

998344 

13 

941952 

2421 

058048 

0 

|   Cosine 

Sine 

|   Cotang. 

I ^ 

I^Ta"*^ 

M. 

_ 

85 

^ 

Degre 

iS 

SINES   AND  TANGENTS.     (5  Degrees.) 


141 


PmT 

Sine 

D.    | 

Cosine 

D. 

Tang. 

1   D. 

Cotang. 

r               . 

J 

8-940296 

2403 

9-998344 

19 

8-941952 

2421 

11058048 

60  ) 

941738 

2394 

998333 

19 

943404 

2413 

056596 

59  ) 

2 

943174 

2387 

998322 

19 

944852 

2405 

055148 

58  S 

\* 

944606 

2379 

998311 

19 

946295 

2397 

053705 

57  ) 

946034 

2371 

998300 

19 

947734 

2390 

052266 

56  ) 

s 

947456 

2363 

998289 

19 

949168 

3282 

050832 

55  ) 

948874 

2355 

998277 

19 

950597 

2374 

049403 

54  ) 

(  7 

950287 

2348 

998266 

19 

952021 

2366 

047979 

53) 

8 

951696 

2340 

998255 

19 

953441 

2360 

046559 

52) 

>  9 

953100 

21532 

998243 

19 

954856 

2351 

045144 

51  ) 

{  10 

954499 

2325 

998232 

19 

956267 

2344 

043733 

50  S 

J  U 

8955894 

2317 

9-998220 

19 

8-957674 

2337 

11042326 

49  j 

(  12 

957284 

2310 

998209 

19 

959075 

2329 

040925 

48/ 

<  13 

958670 

2302 

998197 

19 

960473 

2323 

039527 

47  ) 

S  14 

9G0052 

2295 

998186 

19 

961866 

2314 

038134 

46) 

<  15 

961429 

2288 

998174 

19 

963255 

2307 

036745 

45/ 

(   10 

962801 

2280 

998163 

19 

964G39 

2300 

035361 

44) 

s  17 

964170 

2273 

998151 

19 

966019 

2293 

033981 

43  ) 

<  18 

965534 

2266 

998139 

20 

967394 

2286 

032606 

42) 

(  19 

966893 

2259 

998128 

20 

968766 

2279 

031234 

41  ) 

j  20 

968249 

2252 

998116 

20 

970133 

2271 

029867 

40} 

)  21 

8-969600 

2244 

9-998104 

20 

8-971496 

2265 

11028504 

39  1 

S  22 

970947 

2238 

998092 

20 

972855 

2257 

027145 

38, 
37/ 

S  23 

972289 

2231 

998080 

20 

974209 

2251 

025791 

S  24 

973628 

2224 

998068 

20 

975560 

2244 

024440 

36/ 

(  25 

974902 

2217 

998056 

20 

970906 

2237 

023094 

35/ 

S  26 

976293 

2210 

998044 

20 

978248 

2230 

021752 

34  / 

\  27 

977619 

2203 

998032 

20 

979586 

2223 

020414 

33/ 

(  28 

978941 

2197 

998020 

20 

980921 

2217 

019079 

32/ 

(  29 

980259 

2190 

998008 

20 

982251 

2210 

017749 

31 
30  J 

}  30 

981573 

2183 

997996 

20 

983577 

2204 

016423 

)31 
>  32 

8-982883 

2177 

9-997984 

20 

8-984899 

2197 

11015101 

29  I 

984189 

2170 

997972 

20 

986217 

2191 

013783 

28  ( 

)  33 

985491 

2163 

997959 

20 

987532 

2184 

012468 

27  ( 

)  34 

986789 

2157 

997947 

20 

988842 

2178 

011158 

26  ( 

S  35 

988083 

2150 

997935 

21 

990149 

2171 

009851 

25  ( 

S  36 

989374 

2144 

997922 

21 

991451 

2165 

008549 

24  ( 

<37 
)  38 

990660 

2138 

997910 

21 

992750 

2158 

007250 

23  ( 

991943 

2131 

997897 

21 

994045 

2152 

005955 

22( 

)   39 

993222 

2125 

997885 

21 

995337 

2146 

004663 

21  ( 

S  40 

994497 

2119 

997872 

21 

996624 

2140 

003376 

20  ( 

$41 

8-995768 

2112 

9-997860 

21 

8-997908 

2134 

11-002092 

19  J 

)  42 

997036 

2106 

997847 

21 

999188 

2127 

000812 

18) 

)  43 

998299 

2100 

997835 

21 

000465 

2121 

10-999535 

17  ) 

)  44 

999560 

2094 

997822 

21 

001738 

2115 

998262 

16) 

)  45 

9-000816 

2087 

997809 

21 

003007 

2109 

996993 

15  S 

)  46 

002069 

2082 

997797 

21 

004272 

2103 

995728 

14  S 

>  47 

003318 

2076 

997784 

21 

005534 

2097 

994466 

13  ) 

)  48 

004563 

2070 

997771 

21 

006792 

2091 

993208 

12) 

\  49 

005805 

2064 

997758 

21 

008047 

2085 

991953 

11  ) 

S  50 

007044 

2058 

997745 

21 

009298 

2080 

990702 

10  j 

I  51 

8-008278 

2052 

8-997732 

21 

8010546 

2074 

10-989454 

9! 

)  52 

009510 

2046 

997719 

21 

011790 

2068 

983210 

8/ 

)  53 

010737 

2040 

997706 

21 

013031 

2062 

986969 

T( 

)   54 

011962 

2034 

997693 

22 

014268 

2056 

985732 

6/ 

)  55 

013182 

2029 

997680 

22 

015502 

2051 

984498 

5/ 

/  56 
>57 

014400 

2023 

997667 

22 

016732 

2045 

983268 

4  ) 

015613 

2017 

997654 

22 

017959 

2040 

982041 

3/ 

)  58 

016824 

2012 

997641 

22 

019183 

2033 

980817 

2  ) 

)  59 

018031 

2006 

997628 

22 

020403 

2028 

979597 

1  > 

)  60 

1  019235 

2000 

997614 

22 

021620 

2023 

973380 

0? 

Cosine       I 


I      Cotang. 


Tang.       I     M. 


84  Degrees. 


142 


(6  Degrees.)      A  TABLE    OF    LOGARITHMIC 


n*. 

|    Sine 

I   D. 

|   Cosine 

1   D. 

1   Tang. 

1   D. 

Cotang. 

1 

)  o 

9019235 

2000 

0997614 

22 

9-021620 

2023 

10-978380 

60  ) 

i  i 

020435 

1995 

997601 

22 

022834 

2017 

977166 

59  ) 

)  o 

021632 

1989 

997588 

22 

024044 

2011 

975956 

58  ) 

!  3 

022825 

1984 

997574 

22 

025251 

2006 

974749 

57  ) 

I    4 

024016 

1978 

997561 

22 

026455 

2000 

973545 

56  ) 

/  5 

025203 

1973 

997547 

22 

027655 

1995 

972345 

55  ) 

/  6 

026386 

1967 

997534 

23 

028852 

1990 

971148 

54  ) 

^  7 

027567 

1962 

997520 

23 

030046 

1985 

969954 

53 

S  8 

028744 

1957 

997507 

23 

031237 

1979 

968763 

52  ) 

S    9 

029918 

1951 

997493 

23 

032425 

1974 

967575 

51  ) 

)  W 

031089 

1947 

997480 

23 

033609 

1969 

966391 

50  J 

(  H 

9032257 

1941 

9-997466 

23 

9034791 

1964 

10-965209 

49  ) 

(  12 

033421 

1936 

997452 

23 

035969 

1958 

964031 

48  ) 

<  13 

034582 

1930 

997439 

23 

037144 

1953 

962856 

47  ) 

<  14 

035741 

1925 

997425 

23 

C38316 

1948 

961684 

46  ) 

(  15 

036896 

1920 

997411 

23 

039485 

1943 

960515 

45  ) 

(  16 

038048 

1915 

997397 

23 

040651 

1938 

959349 

44  ) 

(  17 

039197 

1910 

997383 

23 

041813 

1933 

958187 

43  ) 

<  18 

040342 

1905 

997369 

23 

C42973 

1928 

957027 

42  ) 

(  19 

041485 

1899 

997355 

23 

044130 

1923 

955870 

41  ) 

(20 

042625 

1894 

997341 

23 

045284 

1918 

954716 

40  ) 

>21 

9  043762 

1889 

9-997327 

24 

9-046434 

1913 

10-953566 

39  ( 

)  22 

044895 

1884 

997313 

24 

047582 

1908 

952418 

38  ( 

>  23 

046026 

1879 

997299 

24 

048727 

1903 

951273 

37  < 

S  24 

047154 

1875 

997285 

24 

049869 

1898 

950131 

36  ( 

)  25 

048279 

1970 

997271 

24 

051008 

1893 

948992 

35  ( 

)  26 

049400 

1865 

997257 

24 

052144 

1889 

947856 

34  \ 

)  27 

050519 

1860 

997242 

24 

053277 

1884 

946723 

33  I 

>  28 

051635 

1855 

997228 

24 

054407 

1879 

945593 

32  ( 

)  29 

052749 

1850 

997214 

24 

055535 

1874 

944465 

31  I 

j  30 

053859 

1845 

997199 

24 

056659 

1870 

943341 

30  J 

(  31 

054966 

1841 

9997185 

24 

9057781 

1865 

10-942219 

29  j 

I  32 

056071 

1836 

997170 

24 

058900 

1869 

941100 

28  ) 

/  33 

057172 

1831 

997156 

24 

060016 

1855 

939984 

27  ) 

(34 
>  35 

058271 

1827 

997141 

24 

061130 

1851 

938870 

26  ) 

059367 

1822 

997127 

24 

062240 

1846 

937700 

25  ) 

I  36 

060460 

1817 

997112 

24 

063348 

1842 

936652 

24  ) 

/  37 

061551 

1813 

997098 

24 

064453 

1837 

935547 

23  ) 

(  38 

062639 

1808 

997083 

25 

065556 

1833 

934444 

22  ) 

<  39 

063724 

1804 

997068 

25 

066655 

1828 

833345 

21  ) 

(  40 

064806 

1799 

997053 

25 

067752 

1824 

932248 

20  S 

(  41 

9065885 

1794 

9997039 

25 

9-068846 

1819 

10-931154 

19  j 

(  42 

066902 

1790 

997024 

25 

069938 

1815 

930062 

18  ) 

(  43 

068036 

1786 

997009 

25 

071027 

1810 

928973 

17  ) 

<  44 

069107 

1781 

996994 

25 

072113 

1806 

927887 

16  ) 

<  45 

07UI76 

1777 

996979 

25 

073197 

1802 

926803 

15  ) 

(  46 

071242 

1772 

996964 

25 

074278 

1797 

925722 

14  ) 

(  47 

072306 

1768 

996949 

25 

075356 

1793 

924644 

13  ) 

(  48 

073366 

1763 

996934 

25 

076432 

1789 

923568 

12  ) 

(  49 

074424 

1759 

996919 

25 

077505 

1784 

922495 

11  ) 

(   50 

075480 

1755 

996904 

25 

078576 

1780 

921424 

10  ) 

)  51 

9-076533 

1750 

9-996P89 

25 

9079644 

1776 

10920356 

9  ( 

)  52 

077583 

1746 

996874 

25 

080710 

1772 

919290 

8  ( 

)   53 

078631 

1742 

996858 

25 

081773 

1767 

918227 

7  < 

)  54 

079676 

1738 

996843 

25 

082833 

1763 

917167 

6  { 

,>55 

080719 

1733 

996828 

25 

083891 

1759 

916109 

5  ( 

)  56 

081759 

1729 

996812 

26 

084947 

1755 

915053 

4  { 

)   57 

082797 

1725 

996797 

26 

086000 

1751 

914000 

3  { 

58 

083832 

1721 

996782 

26 

087050 

1747 

912950 

2  ( 

)   59 

084864 

1717 

996766 

26 

088098 

1743 

911902 

1  ( 

1  60 

085894 

1713 

996751 

26 

08tl44 

1738 

910856 

0  / 

Sine  |       Cotang.     | 

83  Degrees. 


^Tjuig^  UlLj 


sines    and    tangents.     (7  Degrees.) 


143 


0 

1 

2 

I 

4 
5 


|       Cosine        | 


I        Tanp. 


D.       |      Cotanff. 


9-08.5894 

1713 

9-996751 

26 

9089144 

1738 

10-9108.56 

60/ 

080922 

1709 

996735 

26 

090187 

1734 

909813 

59) 

087947 

1704 

996720 

26 

091228 

1730 

908772 

58) 

088970 

1700 

996704 

26 

092206 

1727 

907734 

57) 

089990 

1696 

996688 

26 

093302 

1722 

906698 

56) 

091008 

1692 

996673 

26 

094336 

1719 

905664 

55) 

092024 

1688 

996657 

26 

095367 

1715 

904633 

34) 

093037 

1684 

996641 

26 

096395 

1711 

903605 

53) 

094047 

1680 

996625 

26 

097422 

1707 

902578 

52) 

095056 

1676 

996610 

26 

098446 

1703 

901554 

51  ) 

096062 

1673 

996594 

26 

099468 

1699 

900532 

50) 

9097065 

1668 

9-996578 

27 

9-100487 

1695 

10899513 

49  < 

098066 

1665 

996562 

27 

101504 

1691 

898496 

48? 

099065 

1661 

996546 

27 

102519 

1687 

897481 

47? 

101)062 

1657 

996530 

27 

103532 

1684 

896468 

46/ 

101056 

1653 

996514 

27 

104542 

1680 

895458 

45? 

102048 

1649 

996498 

27 

105550 

1676 

894450 

44  ? 

103037 

1645 

996482 

27 

106556 

1672 

893444 

43? 

104025 

1641 

996465 

27 

107559 

1669 

892441 

42? 

105010 

1638 

996449 

27 

108560 

1665 

891440 

41  ' 

105992 

1634 

996433 

27 

109559 

1661 

890441 

40? 
39< 

9-106973 

1630 

9-996417 

27 

9110556 

1658 

10-889444 

10T951 

1627 

996400 

27 

111551 

1654 

888449 

38( 

108927 

1623 

996384 

27 

112543 

1650 

887457 

37  ( 

109901 

1619 

996308 

27 

113533 

1616 

886467 

36  < 

110873 

1616 

996351 

27 

114521 

1643 

885479 

35  ( 

111842 

1612 

996335 

27 

115507 

1639 

884493 

34  ( 

112809 

1608 

996318 

27 

116491 

1636 

883509 

33  ( 

113774 

1605 

996302 

28 

117472 

1632 

882528 

32  ( 

114737 

1601 

996285 

28 

118452 

1629 

881548 

31  ( 

115698 

1597 

996209 

28 

119429 

1625 

880571 

30  < 

9110656 

1594 

9-996252 

28 

9120404 

1622 

10-879596 

29  j 

117613 

1590. 

996235 

28 

121377 

1618 

878623 

28) 

11S567 

1587 

996219 

28 

122348 

1615 

877652 

27) 

119519 

1583 

996202 

28 

123317 

1611 

876683 

26) 

120469 

1580 

996185 

28 

124284 

1607 

875716 

25  S 

121417 

1576 

996108 

28 

125249 

1604 

874751 

24  S 

122362 

1573 

996151 

28 

126211 

1601 

873789 

23  J, 

123306 

1569 

996134 

28 

127172 

1597 

872828 

oo  ( 

124248 

1566 

996J17 

28 

128130 

1594 

871870 

21  J 

125187 

1562 

996100 

28 

129087 

1591 

870913 

90 

9126125 

1559 

9-996083 

29 

9-130041 

1587 

10-8J9959 

869006 

19) 

127060 

1556 

996066 

29 

130994 

1584 

18) 

127993 

1552 

996049 

29 

131944 

1581 

868056 

17  ) 

128925 

1549 

996C32 

29 

132893 

1577 

867107 

16  ) 

129854 

1545 

996015 

29 

133839 

1574 

866161 

15) 
14) 

130781 

1542 

995998 

29 

134784 

1571 

865216 

131706 

1539 

995980 

29 

135726 

1567 

864274 

13) 

132630 

1535 

995963 

29 

136607 

1564 

863333 

12) 

133551 

1532 

995946 

29 

137605 

1561 

862395 

11 

134470 

1529 

995928 

29 

138542 

1558 

861458 

10  J 

9135387 

1525 

9-99591 1 

29 

9139476 

1555 

10-860524 

9  I 

136303 

1522 

995894 

29 

140409 

1551 

859591 

8  I 

137216 

1519 

995876 

29 

141340 

1548 

858660 

7< 

138128 

1516 

995859 

29 

142269 

15-15 

857731 

6  I 

139037 

1512 

995841 

29 

143196 

1542 

856804 

5( 

139944 

1509 

995823 

29 

144121 

1539 

855879 

4  ? 

140850 

1506 

995806 

29 

145044 

1535 

854956 

3? 

141754 

1503 

995788 

29 

145966 

1532 

K54034 

2? 

142655 

1500 

995771 

29 

146885 

1529 

853115 

1  ) 

143555 

1496 

995753 

29 

147803 

1526 

&52197 

0  ) 

S2  Degrees. 


144 


(8  Degrees.)     a    TABLE    OF    LOGARITHMIC 


F"mT 

|    Sine 

D. 

Cosine 

I   D. 

Tan?. 

1    D. 

Cotang. 

r^i 

s  o 

9143555 

1496 

9995753 

30 

9-147803 

1526 

10.852197 

60  I 
59  ( 

(  i 

144453 

1493 

995735 

30 

148718 

1523 

851282 

S  2 

145349 

1490 

995717 

30 

149632 

1520 

850368 

58  ( 

S  3 

14(i243 

1487 

995699 

30 

150544 

1517 

849456 

57  / 

\    4 

147136 

1484 

995681 

30 

151454 

1514 

848546 

56  ( 

\  5 

148026 

1481 

995664 

30 

152363 

1511 

847637 

55  I 

(    6 

148915 

1478 

995646 

30 

153209 

1508 

846731 

54  I 

S  7 

149802 

1475 

995628 

30 

154174 

1505 

845826 

53  J 

S  8 

130686 

1472 

995610 

30 

155077 

1502 

844923 

52  I 

S  9 

151569 

1469 

995591 

30 

155978 

1499 

844032 

51  / 

S  10 

152451 

1466 

995573 

30 

156877 

1496 

843123 

50  { 

>  H 

9153330 

1463 

9-995555 

30 

9157775 

1493 

10-842225 

49  I 

)  13 

154208 

14(30 

995537 

30 

158671 

1490 

841329 

48  { 

S  13 

155083 

1457 

995519 

30 

159565 

1487 

840435 

47  ( 

)  I4 

155957 

1454 

995501 

31 

160457 

1481 

839543 

46  ( 

)  I5 

156830 

1451 

995482 

31 

161347 

1481 

838653 

45  I 

S  16 

157700 

1448 

995464 

31 

162236 

1479 

837764 

44  I 

S 17 

158569 

1445 

995446 

31 

163123 

1476 

836877 

43  ( 

)  18 

159435 

1442 

995127 

31 

164008 

1473 

835992 

42  { 

<  19 

160301 

1439 

995409 

31 

164892 

1470 

835108 

41  ( 

j  20 

161164 

1436 

995390 

31 

165774 

1467 

834226 

40 

)  21 

9-162025 

1433 

9-995372 

31 

9-1G6654 

1464 

10-833346 

39  { 

)  22 

102885 

1430 

995353 

31 

167532 

1461 

832468 

38  ( 

)  23 

163743 

1427 

995334 

31 

168409 

1458 

831591 

37  ( 

)  24 

164600 

1424 

995316 

31 

169284 

1455 

830716 

36  < 

)  25 

165454 

1422 

995297 

31 

170157 

1453 

829843 

35  < 

>  20 

106307 

1419 

995278 

31 

1710-29 

1450 

828971 

34  ' 

>  27 

167159 

1416 

995260 

31 

171899 

1447 

828101 

33  ( 

)  28 
$29 

168008 

1413 

995241 

32 

172767 

1444 

827233 

32  < 

168858 

1410 

995222 

32 

173634 

1442 

826366 

31  ( 

J  30 

169702 

1407 

995203 

32 

174499 

1439 

825501 

30  ( 

?  31 

9170547 

1405 

9-995184 

32 

9-1753G2 

1436 

10824638 

29  > 

)  32 

171389 

1402 

995165 

32 

176224 

1433 

823776 

28  ) 

>  33 

172230 

1399 

995146 

32 

177084 

1431 

822916 

27  S 

)  34 

173070 

1396 

995127 

32 

177942 

1428 

822058 

26  > 

)  35 

173908 

1394 

995108 

32 

178799 

1425 

821201 

25  ) 

)  36 

174744 

1391 

995089 

32 

179055 

1423 

820345 

24  ) 

)  37 

175578 

1388 

995070 

32 

180508 

1420 

819492 

23  S 

)  38 

176411 

1388 

995051 

32 

181360 

1417 

818640 

22  S 

)  3!) 

177242 

1383 

995032 

32 

182211 

1415 

817789 

21  ) 

)  40 

178072 

1380 

995013 

32 

183059 

1412 

816941 

20  S 

)   41 

9178900 

1377 

9-994993 

32 

9-183907 

1409 

10-816093 

19  ) 

)  42 

17972G 

1374 

994974 

32 

184752 

1407 

815248 

18  > 

)  43 

180551 

1372 

994955 

32 

185597 

1404 

814403 

17  ) 

44 

181374 

1369 

994935 

32 

186439 

1402 

813561 

16  ) 

;  45 

182196 

1366 

994916 

33 

187280 

1399 

812720 

15  ) 

)  46 

183016 

1364 

994896 

33 

188120 

1396 

811880 

14  ) 

I  47 

183834 

1361 

994877 

33 

188958 

1393 

811042 

13  ) 

)  48 

184651 

1359 

994857 

33 

189794 

1391 

810206 

12  ) 

)  49 

185466 

1356 

994838 

33 

190629 

1389 

809371 

11  ) 

/  50 
)  51 

186280 

1353 

994818 

33 

191462 

1386 

808538 

10  > 

9-187092 

1351 

9-994798 

33 

9-192294 

1384 

10-807706 

9  ( 

1  52 

187903 

1348 

994779 

33 

193124 

1381 

806870 

8  I 

I  53 

188712 

1346 

994759 

33 

193953 

1379 

806047 

7  ( 

}54 

1895J9 

1343 

994739 

33 

194780 

1376 

805220 

6  I 

(  55 

190325 

1341 

994719 

33 

195606 

1374 

804394 

5  I 

I  56 

191130 

1338 

994700 

33 

196430 

1371 

803570 

4  I 

>57 

191933 

1336 

994680 

33 

197253 

1369 

802747 

3  > 

58 

192734 

1333 

9946)50 

33 

198074 

1366 

801926 

2  > 

\  59 

193534 

1330 

994640 

33 

198894 

1364 

801106 

1  ) 

\  60 

194332 

1328 

994620 

33 

199713 

1361 

800287 

0  > 

Cosine   | 

Sine 

.^1 

Cotang-. 

.~^~ 

,^Tan&^ 

^mJ 

81  Degrees. 


SINES    AND     TANGENT! 


(9  Degrees.) 


145 


Tm. 

1    Sine 

1    D. 

|   Cosine 

1  D. 

Tan*. 

1    D. 

Cotanff. 

. 

(  o 

19-194332 

1328 

9-994020 

33 

9 199713 

1361 

10-800287 

60 

(  1 

105129 

1326 

994600 

33 

200529 

1359 

799471 

59 

(  2 

1£5925 

1323 

994580 

33 

201345 

1356 

798655 

58 

(  3 

196719 

1321 

994560 

34 

202159 

1354 

797841 

57  ( 

(    4 

1  197511 

1318 

994540 

34 

202971 

1352 

797029 

56  ( 

(  5 

198302 

1316 

994519 

34 

203782 

1349 

796218 

55? 

(    6 

199091 

1313 

994499 

34 

204592 

1347 

795408 

54? 

S  7 

199879 

1311 

994479 

34 

205400 

1345 

794600 

53? 

<  8 

2(10666 

1308 

994459 

34 

206207 

1342 

793793 

52? 

\  9 

201451 

1306 

994438 

34 

207013 

1340 

792987 

51? 

)  10 

202234 

1304 

994418 

34 

207817 

1338 

792183 

50? 

>  H 

9203017 

1301 

9-994397 

34 

9-208619 

1335 

10-791381 

49  ( 

)  & 

203797 

1299 

994377 

34 

200420 

1333 

790580 

48  ( 

>  13 

204577 

1296 

994357 

34 

210220 

1331 

789780 

47  ( 

>  14 

205354 

1294 

994336 

34 

211018 

1328 

788982 

41)  ( 

>  i5 

206131 

1292 

994316 

34 

211815 

1326 

788185 

45  ( 

S  16 

206906 

1289 

994295 

34 

212611 

1324 

787389 

44  ( 

)  I7 

207679 

1287 

994274 

35 

213405 

1321 

786595 

43  ( 

>  18 

208452 

1285 

994254 

35 

214198 

1319 

785802 

42  { 

)  I9 

209222 

1282 

994233 

35 

214989 

1317 

785011 

41  ( 

S  20 

209992 

1280 

994212 

35 

215780 

1315 

784220 

40  ( 

,1 22 

9-210760 

1278 

9-994191 

35 

9216568 

1312 

10-783432 

39  j 
38  < 

211526 

1275 

994171 

35 

217356 

1310 

782644 

,l  23 

212291 

1273 

994150 

35 

218142 

1308 

781858 

37  S 

)  24 

213055 

1271 

994129 

35 

218926 

1305 

781074 

36  \ 

>  25 

213818 

1268 

994108 

35 

219710 

1303 

780290 

35$ 

)  26 

214579 

1266 

994087 

35 

220492 

1301 

779508 

34  \ 

)  27 

215338 

1264 

994066 

35 

221272 

1299 

778728 

33S 

)  28 

216097 

1261 

994045 

35 

222052 

1297 

777948 

32  ( 

>  29 

216854 

1259 

994024 

35 

222830 

1294 

777170 

31  ( 

j  30 

217609 

1257 

994003 

35 

223606 

1292 

776394 

30  < 

(  31 

9-218363 

1255 

9-993981 

35 

9-224382 

1290 

10-775618 

29) 

(  32 

219116 

1253 

993960 

35 

225156 

1288 

774844 

28) 

(  33 

219868 

1250 

993939 

35 

225929 

1286 

774071 

27) 

S  34 

220018 

1248 

993918 

35 

226700 

1284 

773300 

26  ) 

(  35 

221367 

1246 

993896 

36 

227471 

1281 

772529 

25  > 

{  36 

222115 

1244 

993875 

36 

228239 

1279 

771761 

24  \ 

(  37 

222861 

1242 

993854 

36 

229007 

1277 

770993 

23  S 

(  38 

223606 

1239 

993832 

36 

229773 

1275 

770227 

22  ) 

(  39 

224349 

1237 

993811 

36 

230539 

1273 

769461 

21  ) 

(  40 

225092 

1235 

993789 

36 

231302 

1271 

768698 

20  j 

>  41 

9225833 

1233 

9-993768 

36 

9-232065 

1269 

10-767935 

19.! 

S  42 

226573 

1231 

993746 

36 

232826 

1267 

767174 

18»i 

>  43 

227311 

1228 

993725 

36 

233586 

1265 

766414 

17) 

)  44 

228048 

1226 

993703 

36 

234345 

1262 

765655 

16  ) 

)  45 

228784 

1224 

993681 

36 

235103 

1260 

764897 

15  ) 

)  46 

229518 

1222 

993660 

36 

235859 

1258 

764141 

14  > 
13, 

)  47 

230252 

1220 

993638 

36 

236614 

1256 

763386 

>48 

230984 

1218 

993616 

36 

237368 

1254 

762632 

12 

\49 

231714 

1216 

993594 

37 

238120 

1252 

761880 

11) 

I  50 

232444 

1214 

993572 

37 

238872 

1250 

761128 

10  j 

51 

9-233172 

1212  • 

9-993550 

37 

9-239622 

1248 

10-760378 

9( 

>52 

233899 

1209 

993528 

37 

240371 

1246 

759629 

8  > 

J  53 

234625 

1207 

993506 

37 

241118 

1244 

758882 

7 

>  54 

235349 

1205 

993484 

37 

241865 

1242 

758135 

6 

>  55 

236073 

1203 

993462 

37 

242610 

1240 

757390 

5 

)  56 

236795 

1201 

993440 

37 

243354 

1238 

756646 

4 

)  57 

237515 

1199 

993418 

37 

244097 

1236 

755903 

3 

)  58 

238235 

1197 

993396 

37 

244839 

1234 

755161 

2 

>  59 

238953 

1195 

993374 

37 

245579 

1232 

754421 

1) 

)  60 

239670 

1193 

993351 

37 

246319 

1230 

753681 

° 

L-^ 

Cosine   * 

-*»-^  <^^- 

Sine     | 

Cotanff. 

r J 

^•^1 

51 

80  Degrees. 


tf>~ 


146 


(10 Degrees.;    a   table   of   logarithmic 


M. 

0 

1 
•2 

a 

4 
9 

(i 
7 
8 
B 
10 
11 
19 
j:j 

14 
15 
16 

17 
18 
19 

20 
31 

SS 
S3 
34 

'25 
96 
27 
2d 
20 
30 
31 
32 
33 
34 
35 
36 
37 
38 
30 
40 
41 
42 
43 
.'  44 
45 
4li 
47 
4^ 
41 
50 
51 
(52 
(  53 

<  54 
(  55 

<  50 
(  57 
(  58 

<  51) 
(  60 


Sine 

i   D- 

Cosine 

D. 

Tansf. 

1   D- 

Cotang. 

9239670 

1193 

9-903351 

37 

9-246319 

1230 

10-753681 

60 

240386 

1191 

9J3329 

37 

247057 

1228 

752943 

59 

241101 

1189 

993307 

37 

247794 

1226 

752206 

58 

241814 

1187 

993285 

37 

248530 

1224 

751470 

57 

242526 

1185 

993262 

37 

249264 

1222 

750736 

56 

243237 

1183 

993240 

37 

249998 

1220 

750002 

55 

243947 

1181 

993217 

38 

250730 

1218 

749270 

54 

244656 

1179 

993195 

38 

251461 

1217 

748539 

53 

245363 

1177 

993172 

38 

252191 

1215 

747809 

52 

246069 

1175 

993149 

38 

252920 

1213 

747080 

51 

246775 

1173 

993127 

38 

253648 

1211 

740352 

50 

9-247478 

1171 

9-993104 

38 

9-2.54374 

1209 

10-745626 

49 

248181 

1169 

993081 

38 

255100 

1207 

744900 

48 

248883 

1167 

993059 

38 

255824 

1205 

744176 

47 

24J583 

1165 

993036 

38 

256547 

1203 

743453 

46 

250282 

1163 

903013 

38 

257269 

1201 

742731 

45 

250980 

1161 

992990 

38 

257990 

1200 

742010 

44 

251677 

1159 

902967 

38 

258710 

1198 

741290 

43 

252373 

1158 

992944 

38 

259429 

1196 

740571 

42 

253067 

1156 

902921 

38 

260146 

1194 

739854 

41 

253761 

1154 

992898 

38 

260863 

1192 

739137 

40 

9254453 

1152 

9-902875 

38 

9-261578 

1190 

10-738422 

39 

255144 

1150 

992852 

38 

262292 

1189 

7377u8 

38 

255834 

1148 

992829 

30 

263005 

1187 

736995 

37 

256523 

1146 

902806 

39 

263717 

1185 

736283 

36 

257211 

1144 

9J2783 

39 

264428 

1183 

735572 

35 

257898 

1142 

992759 

39 

265138 

1181 

734802 

34 

258583 

1141 

992736 

39 

265847 

1179 

734153 

33 

259268 

1139 

992713 

39 

260555 

1178 

733445 

32 

259951 

1137 

992690 

39 

267261 

1176 

732739 

31 

260633 

1135 

992o66 

39 

267967 

1174 

732033 

30 

9-261314 

1133 

9-992643 

39 

9-268671 

1172 

10731329 

29 

261994 

1131 

992619 

39 

2U9375 

1170 

730625 

28 

262673 

1130 

992596 

39 

270077 

1169 

729923 

27 

263351 

1128 

992572 

39 

270779 

1167 

729221 

26 

264027 

1126 

992549 

39 

271479 

1165 

728521 

25 

204703 

1124 

992525 

39 

272178 

1164 

727822 

24 

265377 

1122 

992501 

39 

272876 

1162 

727124 

23 

266051 

1120 

9J2478 

40 

273573 

1160 

726427 

22 

266723 

1119 

9J2454 

40 

274269 

1158 

725731 

21 

267395 

1117 

992430 

40 

274i>64 

1157 

725036 

20 

9-268065 

1115 

9-992406 

40 

9-275658 

1155 

10-724342 

19 

268734 

1113 

99;J382 

40 

276351 

1153 

723649 

18 

269402 

1111 

992359 

40 

277043 

1151 

722957 

17 

270069 

1110 

992335 

40 

277734 

1150 

722266 

16 

270735 

1108 

992311 

40 

278424 

1148 

721576 

15 

271400 

1106 

992287 

40 

279113 

1147 

720887 

14  < 

272064 

1105 

992263 

40 

279801 

1145 

720199 

13  < 

272726 

1103 

992239 

40 

280488 

1143 

719512 

12  < 

273388 

1101 

992214 

40 

281174 

1141 

718826 

11 

274049 

1199 

992190 

40 

281858 

1140 

718142 

10  < 

9274708 

1098 

9-992166 

40 

9-282542 

1138 

10-717458 

9 

275367 

1096 

992142 

40 

283225 

1136 

716775 

8( 

276024 

1094 

992117 

41 

283907 

1135 

7160"3 

7 

.  276681 

1092 

992093 

41 

284588 

1133 

715412 

6 

277337 

1001 

992069 

41 

285268 

1131 

714732 

5 

277991 

1689 

992044 

41 

285947 

1130 

714053 

4 

278644 

1087 

992020 

41 

286624 

1128 

713376 

3 

279297 

1086 

991996 

41 

287301 

1126 

712609 

2 

279948 

1084 

991971 

41 

287977 

1125 

712023 

1 

286599 

1082 

991947 

41 

288652 

1123 

711348 

• 

70  Degrees. 


SINES   AND  TANGENTS.     (11  Degrees.) 


U7 


Pm^ 

|    Sins 

1   D- 

Cosine 

D. 

Tang. 

D. 

Cotang. 

; 

i     0 

9-280599 

1082 

9-991947 

41 

9-288652 

1123 

10-711348 

60  J 

I     1 

281248 

1081 

991922 

41 

289326 

1122 

710674 

59  P 

I    2 

281897 

1079 

991897 

41 

289999 

1120 

710001 

58) 

)  3 

282544 

1077 

991873 

41 

290671 

1118 

709329 

57) 

?  4 

283190 

1076 

991848 

41 

291342 

1117 

708658 

56? 

)  5 

283836 

1074 

991823 

41 

292013 

1115 

707987 

55? 

)  6 

284480 

1072 

991799 

41 

292682 

1114 

707318 

54/ 

)  7 

285124 

1071 

991774 

42 

293350 

1112 

706650 

53) 

)  8 

285766 

1069 

991749 

42 

294017 

1111 

705983 

52? 

/  9 

286408 

1067 

991724 

42 

294684 

1109 

705316 

51  ) 

)  10 

287048 

1066 

991699 

42 

295349 

1107 

704051 

50, 

11 

9-287687 

1064 

9-991674 

42 

9-296013 

1106 

10-703987 

49  < 

12 

288326 

1063 

991649 

42 

296677 

1104 

703323 

48  < 

/  13 

288904 

1061 

991624 

42 

297339 

1103 

702661 

47  < 

<  U 

289600 

1059 

991599 

42 

298001 

1101 

701999 

46  ( 

15 

290236 

1058 

991574 

42 

298662 

1100 

701338 

45  ( 

1(5 

290870 

1056 

991549 

42 

299322 

1098 

700678 

44  ( 

17 

291504 

1054 

991524 

42 

299980 

1096 

700020 

43  ( 

18 

292137 

1053 

991498 

42 

300638 

1095 

699302 

42  ( 

>  19 

292768 

1051 

991473 

42 

301295 

1093 

698705 

41  ( 

)  20 

293399 

1050 

991448 

42 

301951 

1092 

698049 

40  ( 

i  21 

9294029 

1048 

9-991422 

42 

9-302607 

1090 

10-697393 

39  ) 

J  22 

(  23 
f  24 

294658 

1046 

991397 

42 

303261 

1089 

690739 

38  S 

295286 

1045 

991372 

43 

303914 

1087 

696086 

37) 

295913 

1043 

991346 

43 

304567 

1086 

695433 

36  S 

'25 

296539 

1042 

991321 

43 

305218 

1084 

694782 

35  ) 

(  2(j 

297164 

1040 

991295 

43 

305869 

1083 

694131 

34  ) 

27 

297788 

1039 

991270 

43 

306519 

1081 

693481 

33 

Hs 

298412 

1037 

991244 

43 

307168 

1080 

092832 

32  \ 

i  20 

299034 

1036 

991218 

43 

307815 

1078 

692185 

31  < 

|  30 

299655 

1034 

991193 

43 

308463 

1077 

691537 

30  J 

!  31 

9-300276 

1032 

9-991167 

43 

9-309109 

1075 

10-690891 

29) 

(  32 

300895 

1031 

991141 

43 

309754 

1074 

690246 

28) 

33 

301514 

1029 

991115 

43 

310398 

1073 

689602 

27) 

34 

302132 

1028 

991090 

43 

311042 

1071 

688958 

26) 

35 

302748 

1026 

991064 

43 

311685 

1070 

688315 

25) 

3fi 

303364 

1025 

991038 

43 

312327 

1068 

687673 

24  ) 

37 

303979 

1023 

991012 

43 

312967 

1067 

687033 

23  > 

38 

304593 

1022 

990986 

43 

313608 

1065 

686392 

22) 

39 

305207 

1020 

990960 

43 

314247 

1064 

685753 

21  ) 

40 

305819 

1019 

990934 

44 

314885 

1062 

685115 

20) 

41 

9-306430 

1017 

9-990908 

44 

9  315523 

1061 

10684477 

19  J 

42 

307041 

1016 

990882 

44 

316159 

1060 

683841 

18  ) 

43 

307650 

1014 

990855 

44 

316795 

1058 

683205 

17  ) 

44 

308259 

1013 

990829 

44 

317430 

1057 

682570 

16) 

45 

308867 

1011 

990803 

44 

318064 

1055 

681936 

15) 

4(5 

309474 

1010 

990777 

44 

318697 

1054 

681303 

14  ) 

47 

310080 

1008 

990750 

44 

319329 

1053 

680671 

13) 

48 

310685 

1C07 

990724 

44 

319961 

1051 

680039 

12) 

49 

311289 

1005 

990697 

44 

320592 

1050 

679408 

11  I 

i  50 

311893 

1004 

990671 

44 

321222 

1048 

678778 

10  ) 

1  51 

9312495 

1003 

9-990644 

44 

9-321851 

1047 

10-678149 

A 

)  52 

313097 

1001 

990618 

44 

322479 

1045 

677521 

8 

53 

313698 

1000 

990591 

44 

323106 

1044 

676894 

7 

54 

314297 

998 

990565 

44 

323733 

1043 

676267 

6/ 

55 

314897 

997 

990538 

44 

324358 

1041 

675642 

5  < 

)  56 

315495 

996 

990511 

45 

324983 

1040 

675017 

4\ 

)  57 

316092 

994 

990485 

45 

325007 

1039 

674393 

3^ 

58 

316689 

993 

990458 

45 

326231 

1037 

673769 

! 

1  59 

317284 

991 

990431 

45 

326853 

1036 

673147 

)  60 

317879 

990  1 

990404  ' 

45 

327475 

1035 

672525 

On 

_L*b> 


148 


(12  Degrees.)      A   TABLE    OF    LOGARITHMIC 


0 

9-317879 

990 

9-990404 

45 

9-327474 

1035 

10672526 

60 

1 

318473 

988 

990378 

45 

328095 

1033 

671905 

59 

2 

319066 

987 

990351 

45 

328715 

1032 

671285 

58 

3 

319G58 

986 

990324 

45 

329334 

1030 

670666 

57 

4 

320249 

984 

990297 

45 

329953 

1029 

670047 

56 

5 

320840 

983 

990270 

45 

330570 

1028 

669430 

55 

6 

321430 

982 

990243 

45 

331187 

1026 

668813 

54 

7 

322019 

980 

990215 

45 

331803 

1025 

668197 

53 

8 

322607 

979 

990188 

45 

332418 

1024 

6G7582 

52 

9 

323194 

977 

990161 

45 

333033 

1023 

666967 

51 

10 

323780 

976 

990134 

45 

333646 

1021 

G66354 

50 

11 

9324366 

975 

9-990107 

46 

9334259 

1020 

10-665741 

49 

12 

324950 

973 

990079 

46 

334871 

1019 

665129 

48 

13 

325534 

972 

990052 

46 

335482 

1017 

664518 

47 

14 

326117 

970 

990025 

46 

336093 

1016 

663907 

46 

15 

326700 

969 

989997 

46 

336702 

1015 

663298 

45 

16 

327281 

968 

989970 

46 

337311 

1013 

662689 

44 

17 

327862 

966 

989942 

46 

337919 

1012 

662081 

43 

18 

328442  * 

965 

989915 

46 

338527 

1011 

661473 

42 

19 

329021 

964 

989887 

46 

339133 

1010 

660867 

41 

20 

329599 

962 

989860 

46 

339739 

1008 

660261 

40 

21 

9330176 

961 

9-989832 

46 

9-340344 

1007 

10659656 

39 

22 

330753 

960 

989804 

46 

340948 

1006 

659052 

38 

23 

331329 

958 

989777 

46 

341552 

1004 

658448 

37 

24 

331903 

957 

989749 

47 

342155 

1003 

657845 

36 

25 

332478 

956 

989721 

47 

342757 

1002 

657243 

35 

26 

333051 

954 

989693 

47 

343358 

1000 

656642 

34 

27 

333624 

953 

989665 

47 

343958 

999 

656042 

33 

28 

334195 

952 

989637 

47 

344558 

998 

655442 

32 

29 

334766 

950 

989609 

47 

345157 

997 

654843 

31 

30 

335337 

949 

989582 

47 

345755 

996 

654245 

30 

31 

9335906 

948 

9-989553 

47 

9-346353 

994 

10-653647 

29 

32 

336475 

946 

989525 

47 

346949 

993 

653051 

28  < 

33 

337043 

945 

989497 

47 

347545 

992 

652455 

27  ' 

34 

337610 

944 

989409 

47 

348141 

991 

651859 

26  ( 

35 

338176 

943 

989441 

47 

348735 

990 

651265 

25  ( 

36 

338742 

941 

989413 

47 

349329 

988 

650671 

24 

37 

339306 

940 

989384 

47 

349922 

987 

650078 

23' 

38 

339871 

939 

989356 

47 

350514 

986 

649486 

22' 

39 

340434 

937 

989328 

47 

351106 

985 

648894 

21  < 

40 

340996 

936 

989300 

47 

351697 

983 

648303 

20  « 

41 

9341558 

935 

9-989271 

47 

9-352287 

982 

10  647713 

19  ! 

42 

342119 

934 

989243 

47 

352876 

981 

647124 

18 

43 

342679 

932 

989214 

47 

353465 

980 

646535 

17 ; 

44 

343239 

931 

989186 

47 

354053 

979 

645947 

16 

45 

343797 

930 

989157 

47 

354640 

977 

645360 

15 

46 

344355 

929 

989128 

48 

355227 

976 

644773 

14 

47 

344912 

927 

989100 

48 

355813 

975 

644187 

13) 

48 

345469 

926 

989071 

48 

356398 

974 

643602 

12  S 

49 

346024 

925 

989042 

48 

356982 

973 

643018 

"  J 

50 

346579 

924 

989014 

48 

357566 

971 

642434 

10  s 

51 

9-347134 

922 

9-988985 

48 

9-358149 

970 

10-641851 

9? 

52 

347687 

921 

988956 

48 

358731 

969 

641269 

6f 

53 

348240 

920 

988927 

48 

359313 

968 

640687 

?? 

54 

348792 

919 

988898 

48 

359893 

967 

640107 

6? 

55 

349343 

917 

988869 

48 

360474 

966 

639526 

5) 

5G 

349893 

916 

988840 

48 

361053 

965 

638947 

4/ 

57 

350443 

915 

988811 

49 

361632 

963 

638368 

3? 

58 

350992 
351540 

914 

988782 

49 

362210 

962 

637790 

2> 

59 

913 

988753 

49 

362787 

961 

637213 

1 

60 

352088 

911 

988724 

49 

363364 

960 

636636 

° 

|  M. 


77  Degrees. 


SINES   AND   tangents.    (13  Degrees.) 


D.       I      Cosine      I 


Tang.       | 


Cotang.       | 


{  o 

9-352088 

911 

9-988724 

49 

9363364 

960 

10-636036 

60) 

S  i 

352635 

910 

98S695 

49 

363940 

959 

636060 

59  ) 

\   2 

353181 

909 

988666 

49 

364515 

958 

635485 

58  ) 

S  3 

353726 

908 

988636 

49 

365090 

957 

634910 

57  ) 

354271 

907 

988607 

49 

365664 

955 

634336 

56) 

354815 

905 

988578 

49 

366237 

954 

633763 

55  ) 

355358 

904 

988548 

49 

366810 

953 

633190 

54  ) 

1  7 

355901 

903 

988519 

49 

367382 

952 

632618 

53) 

I    8 

356443 

902 

988489 

49 

367953 

951 

632047 

52) 

I  9 

356984 

901 

988460 

49 

368524 

950 

631476 

51  ) 

MO 

357524 

899 

988430 

49 

369094 

949 

630906 

50  \ 

)  U 

9358064 

898 

9-988401 

49 

9-369663 

948 

10-630337 

49) 

)  12 

358603 

897 

988371 

49 

370232 

946 

629768 

48/ 

)  I3 

359141 

896 

988342 

49 

370799 

945 

629201 

47  > 

(  14 

359(578 

895 

988312 

50 

371367 

944 

628633 

46  ) 

(  15 

360215 

893 

988282 

50 

371933 

943 

628067 

45  ) 

<  16 

360752 

892 

988252 

50 

372499 

942 

627501 

44  ) 

(  17 

361287 

891 

988223 

50 

373064 

941 

626936 

43  ) 

(  18 

361822 

890 

988193 

50 

373029 

940 

626371 

42  ) 

<  19 

362356 

889 

988163 

50 

374193 

939 

625807 

41  ) 

(20 

362889 

888 

988133 

50 

374756 

938 

625244 

40  ) 

)21 

9363422 

887 

9-988103 

50 

9-375319 

937 

10-624681 

39  ( 

)22 

363954 

885 

988073 

50 

375881 

935 

624119 

38  ( 

S  23 

364485 

884 

988043 

50 

376442 

934 

623558 

37  < 

>24 
(25 

365016 

883 

988013 

50 

377003 

933 

622997 

36  < 

365546 

882 

987983 

50 

377563 

932 

622437 

35  < 

>26 
(27 

,28 

J  29 
\30 

366075 

881 

987953 

50 

378122 

931 

621878 

34  < 

366604 

880 

987922 

50 

378681 

930 

621319 

33  ( 

367131 

879 

987892 

50 

379239 

929 

620761 

32  ( 

367659 

877 

987862 

50 

379797 

928 

620203 

31  ( 

368185 

876 

987832 

51 

380354 

927 

619646 

30/ 

>  31 

9-368711 

875 

9-987801 

51 

9-380910 

926 

10-619090 

29( 

>32 

369236 

874 

987771 

51 

381466 

925 

618534 

28  < 

>33 

369761 

873 

987740 

51 

382020 

924 

617980 

27  < 

>34 

370285 

872 

987710 

51 

382575 

923 

617425 

26  < 

>35 

370808 

871 

987679 

51 

383129 

922 

616871 

25  ( 

>36 

371330 

870 

987649 

51 

383682 

921 

616318 

24  < 

>37 

371852 

869 

987618 

51 

384234 

920 

615766 

23  < 

>38 

372373 

867 

987588 

51 

384786 

919 

615214 

22< 

>39 

372894 

866 

987557 

51 

385337 

918 

614663 

21  ( 

>40 

373414 

865 

987526 

51 

385888 

917 

614112 

20  < 

41 

9-373933 

864 

9-987496 

51 

9-386438 

915 

10-613562 

19  / 

(42 

374452 

863 

987465 

51 

386987 

914 

613013 

18  ) 

?43 

374970 

862 

987434 

51 

387536 

913 

612464 

17  ) 

<44 

375487 

861 

987403 

52 

388084 

912 

611916 

16  S 

?45 

376003 

860 

987372 

52 

388631 

911 

611369 

15  > 

(46 

376519 

859 

987341 

52 

389178 

910 

610822 

14  ) 

?47 

377035 

858 

987310 

52 

389724 

909 

610276 

13  ) 

?48 

377549 

857 

987279 

52 

390270 

908 

609730 

12 '» 

?49 

378063 

856 

987248 

52 

390815 

907 

609185 

11  (i 

?50 

378577 

854 

987217 

52 

391360 

906 

608640 

10 

(51 

9-379089 

853 

9-987186 

52 

9-391903 

905 

10-608097 

9  i1 

(52 

379601 

852 

987155 

52 

392447 

904 

607553 

8  , 

(53 

380113 

851 

987124 

52 

392989 

903 

607011 

1  i1 

J54 

380624 

850 

987092 

52 

393531 

902 

606469 

6, 
3, 

(55 

381134 

849 

987061 

52 

394073 

901 

605927 

56 

381643 

848 

987030 

52 

394614 

900 

605386 

?57 

382152 

847 

986998 

52 

395154 

899 

604846 

(58 

382661 

846 

986967 

52 

395694 

898 

604306 

(59 

383108 

845 

986936 

52 

396233 

897 

603767 

(60 

383675 

844 

986904 

52 

396771 

896 

603229 

° 

l~ 

Cosine 

Sine 

Cotan|. 

I 1 

^Tang^j 

^j 

76  Degreea 


150 

(14  Degree 

s.)  A  TABLE 

OP  LOGARITHMIC 

^ST 

Sine 

1   »• 

|   Cosme 

1   D. 

1   Tang. 

1   D. 

Cotang. 

\ 

(   ° 

9383675 

844 

9-986904 

52 

9396771 

896 

10-603229 

60 

,  l 

384182 

843 

986873 

53 

397309 

896 

602691 

59 

.  2 

384687 

842 

986841 

53 

397846 

895 

602154 

58 

3 

385192 

841 

986809 

53 

398383 

894 

601617 

57 

.  4 

385697 

840 

986778 

53 

398919 

893 

601081 

56 

.  5 

386201 

839 

986746 

53 

399455 

892 

600545 

55 

6 

386704 

838 

986714 

53 

399990 

891 

600010 

54 

>  7 

387207 

837 

986683 

53 

400524 

890 

599476 

53 

8 

387709 

836 

986651 

53 

401058 

889 

598942 

52 

9 

388210 

835 

986619 

53 

401591 

888 

598409 

51 

1  10 

388711 

834 

986587 

53 

402124 

887 

597876 

50 

11 

9389211 

833 

9986555 

53 

9-402656 

886 

10*597344 

49 

12 

389711 

832 

986523 

53 

403187 

885 

596813 

48 

13 

390210 

831 

986491 

53 

403718 

884 

596282 

47 

14 

390708 

830 

986459 

53 

404249 

883 

595751 

46 

15 

391206 

828 

986427 

53 

404778 

882 

595222 

45 

16 

391703 

827 

986395 

53 

405308 

881 

594092 

44 

17 

392199 

826 

986303 

54 

405836 

880 

594164 

43 

18 

392695 

825 

986331 

54 

406364 

879 

593636 

42 

19 

393191 

824 

986299 

54 

406892 

878 

593108 

41 

20 

393685 

823 

986266 

54 

407419 

877 

592581 

40 

21 

9394179 

822 

9-986234 

54 

9-407945 

876 

10-592055 

39 

22 

394673 

821 

986202 

54 

408471 

875 

591529 

38 

23 

395166 

820 

986169 

54 

408997 

874 

591003 

37 

24 

395658 

819 

986137 

54 

409521 

874 

590479 

36 

25 

390150 

818 

986104 

54 

410045 

873 

589955 

35 

26 

396641 

817 

986072 

54 

410569 

872 

589431 

34 

27 

397132 

817 

986039 

54 

411092 

871 

588908 

33 

28 

397621 

816 

986007 

54 

411615 

870 

588385 

32 

29 

398111 

815 

985974 

54 

412137 

869 

587863 

31 

30 

398600 

814 

985942 

54 

412658 

868 

587342 

30 

31 

9-399088 

813 

9-985909 

55 

9-413179 

867 

10-586821 

29 

32 

399575 

812 

98.5876 

55 

413699 

866 

586301 

28 

33 

400062 

811 

985843 

55 

414219 

8C5 

585781 

27 

34 

400549 

810 

985811 

55 

414738 

864 

585262 

26 

35 

401035 

809 

985778 

55 

415257 

864 

584743 

25 

36 

401520 

808 

985745 

55 

415775 

863 

584225 

24 

37 

402005 

807 

985712 

55 

416293 

862 

583707 

23 

38 

402489 

806 

985679 

55 

416810 

861 

583190 

22  < 

39 

402972 

805 

985646 

55 

417326 

860 

582674 

21 

40 

403455 

804 

985613 

55 

417842 

859 

582158 

20 

41 

9-403938 

803 

9-985580 

55 

9-418358 

858 

10581642 

19  j 

42 

404420 

802 

985547 

55 

418873 

857 

581127 

18 

43 

404901 

801 

985514 

55 

419387 

856 

580613 

17 

44 

405382 

800 

985480 

55 

419901 

855 

580099 

16 

45 

405862 

799 

985447 

55 

420415 

855 

579585 

15  < 

46 

406341 

798 

985414 

56 

420927 

854 

579073 

14  ) 

47 

406820 

797 

985380 

56 

421440 

853 

578560 

13 

48 

407299 

796 

985347 

56 

421952 

852 

578048 

12 

49 

407777 

795 

985314 

56 

422463 

851 

577537 

11 

50 

408254 

794 

985280 

56 

422974 

850 

577026 

10 

51 

9-408731 

794 

9-985247 

56 

9-423484 

849 

10-576516 

»! 

52 

409207 

793 

985213 

56 

423993 

848 

576007 

8 

53 

409682 

792 

985180 

56 

424503 

848 

575497 

7  ) 

54 

410157 

791 

985146 

56 

425011 

847 

574989 

6  J 

55 

410632 

790 

985113 

56 

425519 

846 

574481 

5  J 

56 

411106 

789 

985079 

56 

426027 

845 

573973 

4  5 

57 

411579 

788 

985045 

56 

426534 

844 

573466 

3 ) 

58 

412052 

787 

985011 

56 

427041 

843 

572959 

2 

59 

412524 

786 

984978 

56 

427547 

843 

572453 

1 

60 

412996 

785 

984944 

56 

428052 

842 

571948 

0  J 

L~^ 

Cosin* 

^^^J 

Sine   | 

I 

Cotang. 

! 

^Tang^J 

M.  , 

75 

Degre 

es. 

AND    TANGENTS.     tl5  De0rees.) 


151 


pr 

|    Sine 

D. 

Cosine 

D. 

|    Tane. 

i   D. 

Cotang. 

1    i 

)  o 

9412996 

785 

9-984944 

57 

9-428052 

842 

10571948 

1  60  < 

413467 

784 

984910 

57 

428557 

841 

571443 

59  I 

i  2 

413938 

783 

984876 

57 

429002 

840 

570938 

58  ( 

57 

56 

)  3 

414408 

783 

984842 

57 

429506 

839 

570434 

)  4 

414878 

782 

984808 

57 

430070 

838 

569930 

i  5 

415347 

781 

984774 

57 

4:'0573 

838 

569427 

55  ( 

<  6 

415815 

780 

984740 

57 

431075 

837 

568925 

54  ( 

S  7 

416283 

779 

98470(3 

57 

431577 

836 

568423 

53  I 

(  8 

416751 

778 

984672 

57 

432079 

835 

567921 

52  I 

(  9 

417217 

777 

984637 

57 

432580 

834 

567420 

51  ( 

<10 

417684 

776 

984603 

57 

433080 

833 

566920 

50  j 

i  n 

9418150 

775 

9-984569 

57 

9-433580 

832 

10'5664SO 

49  < 

)   12 

418015 

774 

984535 

57 

434080 

832 

5(55920 

48  ) 

)  13 

419079 

773 

984500 

57 

434579 

831 

565421 

47  ( 

?  u 

419544 

773 

984466 

57 

435078 

830 

564922 

46  ( 

)   15 

420007 

772 

984432 

58 

435576 

829 

564424 

45  ( 

>  1G 

420470 

771 

984297 

58 

436073 

828 

£63927 

44  ( 

)  17 

420933 

770 

984363 

58 

436570 

828 

563430 

43  ( 

S  18 

421395 

769 

984328 

58 

437067 

827 

562933 

42  ( 

\   19 

421857 

768 

984294 

58 

437563 

826 

562437 

41  ( 

j  20 

422318 

767 

984259 

58 

438059 

825 

561941 

40  ( 

J  21 

9422778 

767 

9-984224 

58 

9-438554 

824 

10-561446 

39  J 

)  22 

423238 

766 

984190 

58 

439048 

823 

5G0952 

38  ) 

)  23 

423697 

765 

984155 

58 

439543 

823 

560457 

37  S 

/  24 

424156 

764 

984120 

58 

440036 

822 

559964 

36  ) 

)  25 

424615 

763 

984085 

58 

440529 

821 

559471 

35  ) 

)  26 

425073 

762 

984050 

58 

441022 

820 

558978 

34  S 

)  27 

425530 

761 

984015 

58 

441514 

819 

558486 

33  ) 

)  28 

425987 

760 

983981 

58 

442006 

819 

557994 

32  \ 

}  29 

426443 

760 

983946 

58 

442497 

818 

557503 

31  \ 

)  30 

426899 

759 

983911 

58 

442988 

817 

557012 

30  < 

1  31 

9427354 

758 

9-983875 

58 

9-443479 

816 

10-556521 

29  j 

(  32 

427809 

757 

983840 

59 

443968 

816 

556032 

28  ) 

I  33 

428263 

756 

983805 

59 

444458 

815 

555542 

27  ) 

(  34 

428717 

755 

983770 

59 

444947 

814 

555053 

26  ) 

/  35 

429170 

754 

983735 

59 

445435 

813 

554565 

25  ) 

(  36 
)  37 

429623 

753 

983700 

59 

445923 

812 

554077 

24  ) 

430075 

752 

983664 

59 

446411 

812 

553.589 

23  ) 

I  38 
)  39 

430527 

752 

983629 

59 

446898 

811 

553102 

22  ) 

430978 

751 

983594 

59 

447384 

810 

552616 

21  ) 

MO 

431429 

750 

983558 

59 

447870 

809 

552130 

20  J 

(41 

9-431879 

749 

9983523 

59 

9-448356 

809 

10-551644 

19  \ 

<42 

432329 

749 

983487 

59 

448841 

808 

551159 

18  ) 

<43 

432778 

748 

983452 

59 

449326 

807 

550674 

I7  ( 

(  44 

433226 

747 

983416 

59 

449810 

806 

550190 

16  ) 

<  45 

433675 

746 

983381 

59 

450294 

806 

549706 

15  ) 

I  46 

434122 

745 

983345 

59 

4.50777 

805 

549223 

14  ) 

<  47 

434569 

744 

983309 

59 

451260 

804 

548740 

13  ; 

(4V 

435016 

744 

983273 

60 

451743 

803 

548257 

12  I 
10  ) 

?49 

435462 

743 

983238 

60 

452225 

802 

547775 

(5(. 

435908 

742 

983202 

60 

452706 

802 

547294 

'  51 

9-436353 

741 

9-983166 

60 

9-453187 

801 

10-546813 

9  ( 

(  52 

436798 

740 

983130 

60 

453668 

800 

546332 

8  ( 

(53 

437242 

740 

983094 

60 

454148 

799 

545852 

7  ( 

v  54 

437686 

739 

983058 

60 

454628 

799 

545372 

6  ( 

i  55 

438129 

738 

983022 

60 

455107 

798 

544893 

5  ( 

<■  56 

438572 

737 

982986 

60 

455586 

797 

544414 

4  ( 

(  57 

439014 

736 

982950 

60 

456064 

796 

543936 

3  ( 

I  58 

.  439456 

736 

982914 

60 

456542 

796 

543458 

2  ) 

(59 

439897 

735 

982878 

60 

457019 

795 

542981 

1  ) 

(60 

440338 

734 

982842 

60 

457496 

794 

542504 

0  > 

|         Cosine       | 


74  Degrees. 


152 


(16  Degrees.)    a  table  of  logaritmic 


J  ° 

Sine 
9440338  " 

1   D- 
734 

|   Cosine 

|   D. 

1   Tang. 

|   D.   |   Cotang. 

1 

9-982842 

60 

9-457496 

794 

10-542504 

60 

)   1 

440778 

733 

982805 

60 

457973 

793 

542027 

59 

)  2 

441218 

732 

982769 

61 

458449 

793 

541551 

58 

)  3 

441658 

731 

982733 

61 

458925 

792 

541075 

57 

)  4 

442096 

731 

982696 

61 

459400 

791 

540600 

56 

)  5 

442535 

730 

982660 

61 

459875 

790 

540125 

55 

)    6 

442973 

729 

982624 

61 

460349 

790 

539651 

54 

)    7 

443410 

728 

982587 

61 

460823 

789 

539177 

53 

)    8 

443847 

727 

982551 

61 

461297 

788 

538703 

52 

)    9 

444284 

727 

982514 

61 

461770 

788 

538230 

51 

)   10 

444720 

726 

982477 

61 

462242 

787 

537758 

50 

i  U 

9445155 

725 

9-982441 

61 

9-462714 

786 

10537286 

49 

)  12 

445590 

724 

982404 

61 

463186 

785 

536814 

48 

)  13 

446025 

723 

982367 

61 

463658 

785 

536342 

47 

)  14 

446459 

723 

982331 

61 

464129 

784 

535871 

46 

)  15 

446893 

722 

982294 

61 

464599 

783 

535401 

45 

)  I6 

447326 

721 

982257 

61 

465069 

783 

534931 

44 

>  I7 

447759 

720 

9*2220 

62 

465539 

782 

534461 

43 

)  18 

448191 

720 

9132183 

62 

466008 

781 

533992 

12 

)  I9 

448623 

719 

982146 

62 

466476 

780 

533524 

41 

J  20 

449054 

718 

982109 

62 

466945 

780 

533055 

40 

J  21 

9-449485 

717 

9-982072 

62 

9-467413 

779 

10-532587 

39 

)  22 

449915 

716 

982035 

62 

467880 

778 

532120 

38 

)  23 

450345 

716 

981998 

62 

468347 

778 

531653 

37 

)  24 

450775 

715 

981961 

62 

468814 

777 

531186 

36 

)  25 

451204 

714 

981924 

62 

469280 

776 

530720 

35 

)  26 

451G32 

713 

981886 

62 

469746 

775 

530254 

34 

)  27 

452060 

713 

981849 

62 

470211 

775 

529789 

33 

>  28 

452488 

712 

981812 

62 

470676 

774 

529324 

32 

)  29 

452915 

711 

981774 

62 

471141 

773 

528859 

31 

J  30 

453342 

710 

981737 

62 

471605 

773 

528395 

30 

1  31 

9-453768 

710 

9-981699 

63 

9-472068 

772 

10-527932 

29 

>  32 

454194 

709 

981662 

63 

472532 

771 

527468 

28 

)  33 

454619 

708 

981625 

63 

472995 

771 

527005 

27 

)  34 

455044 

707 

981587 

63 

473457 

770 

526543 

28 

}  35 

455469 

707 

981549 

63 

473919 

769 

526081 

25 

)  36 

455893 

706 

981512 

63 

474381 

769 

525619 

24 

)37 

456316 

705 

981474 

63 

474842 

768 

525158 

23 

)  38 
)  39 

456739 

704 

981438 

63 

475303 

767 

524697 

22 

457162 

704 

981399 

63 

475763 

767 

524237 

21 

j  40 

457584 

703 

981361 

63 

476223 

766 

523777 

20 

)  41 

9-458006 

702 

9-981323 

63 

9-476683 

765 

10-523317 

19  - 

)  42 

458427 

701 

981285 

63 

477142 

765 

522858 

18  { 

)  43 

458848 

701 

981247 

63 

477601 

764 

522399 

17 

)  44 

459268 

700 

981209 

63 

478059 

763 

521941 

16 

W5 

459688 

699 

981171 

63 

478517 

763 

521483 

15 

)   46 

460108 

698 

981133 

64 

478975 

762 

521025 

14 

\   47 

460527 

698 

981095 

64 

479432 

761 

520568 

13 

>  48 

460946 

697 

981057 

64 

479889 

761 

520111 

12  < 

v  49 

461364 

696 

981019 

64 

480345 

760 

519655 

11 

J  50 

461782 

695 

980981 

64 

480801 

759 

519199 

It 

)  51 

9-462199 

695 

9-980942 

64 

9-481257 

759 

10  518743 

9 

>  52 

462616 

694 

980904 

64 

481712 

758 

518288 

8< 

;  53 
54 

463032 

693 

980866 

64 

482167 

757 

517833 

7  I 

463448 

693 

980827 

64 

482621 

757 

517379 

6 

.  55 

56 

{   57 

463864 

692 

980789 

64 

483075 

756 

516925 

5 

464279 

691 

980750 

64 

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73  Degrees 


G 

1NES 

AND  TANGENTS.   (17  Degrees.) 

L53 

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Sine    | 

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D. 

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!  ° 

9-465935  1 

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I   51 

9-486467 

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9-978574 

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0 

|        Cosine       | 


|        Sine  | 


Cotang.      | 


Tang. 


72  Degrees 


154 


(18  Degrees.;     a    tabu;    OF    logarithmic 


(  M.  | 

Sine    ! 

D.   | 

Cosine   | 

D.  | 

Tang.    | 

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( 

0 

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9-978206 

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9-977752 

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10-483516 

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69 

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495772 

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518610 

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Cosine         | 


Sine         | 


|       Cotang.      | 


I         Tang. 


J^l 


71  Degrees. 


SINES    AND    TANGENTS.     (19  degrees.) 


155 


r*^"r 

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D.   | 

Cosine 

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Tanf?. 

D. 

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609 

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681 

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608 

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73 

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681 

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608 

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73 

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460163 

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515566 

607 

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73 

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680 

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52  ( 

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73 

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S27 

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S28 

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670 

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669 

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J  30 

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9-530915 

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9-973398 

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9-557517 

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10-442483 

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973169 

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0  ( 

L. 

|   Cosine 



|    Sine 

J 

|   Cotang. 

I 

^Jang^^ 

^M.j 

70  Dejn-ees. 


156 


(20  Degrees.)     A    TABLE    OP    LOGARITHMIC 


M.  | 
0 

1 
2 
3 
4 
5 
G 
7 


2ii 

24 

25 

<,  2(5 

27 

23 

(  29 

J  30 

31 

)   32 

*  33 

34 

)  35 

)   30 

)   37 

S  38 

S  39 

j  40 

j  41 

>  42 

)  43 
)  44 

>  45 
;  46 

47 
1 48 
)  49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


I   D. 


|   D. 


Tang.    |   D.   |   Cotang.   | 


9-534052 

578 

9-972986 

77 

534399 

577 

972l)40 

77 

$34745 

577 

972894 

77 

535092 

577 

972848 

77 

535438 

576 

972802 

77 

535783 

576 

972755 

77 

530129 

575 

972709 

77 

530474 

574 

972663 

77 

53G818 

574 

972617 

77 

5371G3 

573 

972570 

77 

537507 

573 

972524 

77 

9-537851 

572 

9-972478 

77 

538194 

572 

972431 

78 

538538 

571 

972385 

78 

538880 

571 

972338 

78 

539223 

570 

972291 

78 

539505 

570  . 

972245 

78 

539907 

5G9 

972198 

78 

540249 

569 

972151 

78 

540590 

5C8 

972105 

78 

540931 

508 

972058 

78 

9541272 

567 

9-972011 

78 

541G13 

507 

971904 

78 

541953 

506 

971917 

78 

542293 

566 

971870 

78 

542G32 

565 

971823 

78 

542971 

565 

971770 

78 

543310 

564 

971729 

79 

543G49 

564 

971682 

79 

543987 

5G3 

971635 

79 

544325 

5G3 

971588 

79 

9544663 

562 

9-971540 

79 

545000 

562 

971493 

79 

545338 

561 

971446 

79 

545674 

561 

971398 

79 

546011 

560 

971351 

79 

54G347 

560 

971303 

79 

54GG83 

559 

971256 

79 

547019 

559 

971208 

79 

547354 

558 

971 1G1 

79 

547089 

558 

971113 

79 

9548024 

557 

9-971066 

80 

548359 

557 

971018 

80 

543693 

556 

970970 

80 

549027 

556 

970922 

80 

549360 

555 

970874 

80 

549693 

555 

970827 

80 

550020 

554 

970779 

80 

550359 

554 

97073K 

80 

550G92 

553 

970683 

80 

551024 

553 

970635 

80 

9-551356 

552 

9-970580 

80 

551087 

552 

970538 

80 

552018 

5o2 

970490 

80 

552349 

551 

970442 

80 

552080 

551 

970394 

80 

553010 

550 

970345 

81 

553341 

550 

970297 

81 

553070 

549 

970249 

81 

554000 

549 

970200 

81 

554329 

548 

970152 

81 

9-561066 
501459 
561851 
562244 
562636 
563028 
5G3419 
563811 
5642G2 
564592 
564983 

9565373 
5G5763 
566153 
566542 
566932 
567320 
567709 
568098 
5G8486 
568873 

9-569261 
569648 
570035 
570422 
570809 
571195 
571581 
571967 
572352 
572738 

9-573123 
573507 
573892 
574276 
574660 
575044 
575427 
575810 
576193 
576576 

9-576958 
577341 
577723 
578104 
578480 
578867 
579248 
579629 
580009 
580389 

9-580769 
581149 
581528 
581907 
582286 
582G65 
583043 
583422 
583800 
584177 


655 
654 
654 
653 
653 
653 
C52 
652 
651 
651 
650 
650 
649 
649 
649 
648 
648 
G47 
647 
646 
646 
645 
645 
645 
644 
644 
643 
643 
642 
642 
642 
641 
641 
640 
640 
639 
639 
639 
638 
638 
637 
637 
636 
636 
636 
635 
635 
634 
634 
634 
633 
633 
632 
632 
632 
631 
631 
630 
630 
629 


10-438934 
438541 
438149 
437756 
437364 
436972 
436581 
436189 
435798 
435408 
435017 

10-434627 
434237 
433847 
433458 
433068 
432680 
432291 
431902 
431514 
431127 

10-430739 
430352 
429965 
429578 
429191 
428805 
428419 
428033 
427648 
427262 

10-426877 
426493 
426108 
425724 
425340 
424956 
424573 
424190 
423807 
423424 

10-423041 
422659- 
422277 
421896 
421514 
421133 
420752 
420371 
410991 
419611 

10-419231 
418851 
418472 
418093 
417714 
417335 
416957 
416578 
41G200 
415823 


59 

58 

57 

50, 

55  > 

54/ 

53) 

52) 

51) 

50  ) 

49 

48 

47 

46? 

45 

44 

43 

42 

41 

40 

39 

38 

37 

86 

35 

34 

33 

32 

31 

30 

2<) 

88 

27 

36 

25 

'24 

23 

2-2 

21 

20 

19 

18 

17 

1G 

15 

14 

13 

12, 

11 

10 

9 

8 

7 

G 

5 

4< 

fi 


Degrees. 


ai 


Tanjj^jJjLA 


SINES    AND    tangents.     (21  Degrees.) 


157 


n. 

Sine 

D. 

Cosine 

D. 

Tan?. 

D. 

Cotang. 

0 

9-554329 

548 

9970152 

81 

9-584177 

629 

10-415823 

60 

l 

554658 

548 

970103 

81 

584555 

629 

415445 

59 

2 

554987 

547 

960055 

81 

584932 

628 

415068 

58 

3 

555315 

547 

970006 

81 

585309 

628 

414691 

57 

4 

555643 

546 

969957 

81 

585686 

627 

414314 

56 

5 

555971 

546 

969909 

81 

586062 

627 

413938 

55 

6 

556299 

545 

969860 

81 

586439 

627 

413561 

54 

7 

556626 

545 

969811 

81 

586815 

026 

413185 

53 

8 

556953 

544 

969762 

81 

587190 

626 

412810 

52 

9 

557280 

544 

969714 

81 

587566 

625 

412434 

51 

10 

557606 

543 

969665 

81 

587941 

625 

412059 

50 

11 

2  557932 

543 

9-969616 

82 

9588316 

625 

10-411684 

49 

12 

558258 

543 

969567 

82 

588691 

624 

411309 

48 

13 

558583 

542 

969518 

82 

589066 

624 

410934 

47 

14 

558909 

542 

969469 

82 

589440 

623 

410560 

46 

15 

559234 

541 

969420 

82 

589814 

623 

410186 

45 

10 

559558 

541 

969370 

82 

590188 

623 

409812 

44 

17 

559883 

540 

969321 

82 

590562 

622 

409438 

43 

18 

560207 

540 

969272 

82 

590935 

622 

409065 

49 

19 

5C0531 

539 

969223 

82 

591308 

622 

408692 

41 

20 

560855 

539 

969173 

82 

591681 

621 

408319 

40 

21 

9561178 

538 

9-969124 

82 

9-592054 

621 

10-407946 

39 

22 

561501 

538 

969075 

82 

592426 

620 

407574 

38 

23 

561824 

537 

969025 

82 

592798 

620 

407202 

37 

24 

562146 

537 

968976 

82 

593170 

619 

406829 

30 

25 

562468 

536 

968926 

83 

593542 

619 

406458 

35 

20 

562790 

536 

968877 

83 

593914 

618 

406086 

34 

27 

563112 

536 

968827 

83 

594285 

618 

405715 

33 

2? 

563433 

535 

968777 

83 

594656 

618 

405344 

33 

29 

563755 

535 

968728 

83 

595027 

617 

404973 

31 

30 

564075 

534 

968678 

83 

595398 

617 

404602 

30 

31 

9-564396 

534 

9-968628 

83 

9-595768 

617 

10-404232 

29 

32 

564716 

533 

968578 

83 

596138 

616 

403862 

88 

33 

565036 

533 

968528 

83 

596508 

616 

403492 

27 

34 

565356 

532 

968479 

83 

596878 

616 

403122 

26 

35 

565676 

532 

968429 

83 

597247 

615 

402753 

25 

36 

565995 

531 

968379 

83 

597616 

615 

402384 

24 

37 

566314 

531 

968329 

83 

597985 

615 

402015 

23 

38 

566632 

531 

968278 

83 

598354 

614 

401646 

83 

39 

566951 

530 

968228 

84 

598722 

614 

401278 

21 

40 

567269 

530 

968178 

84 

599091 

613 

400909 

20 

41 

9-567587 

529 

9-968128 

84 

9-599459 

613 

10-400541 

19 

42 

567904 

529 

968078 

84 

599827 

613 

400173 

18 

43 

568222 

528 

968027 

84 

600194 

612 

399806 

17 

44 

568539 

528 

967977 

84 

600562 

612 

399438 

16 

45 

568856 

528 

967927 

84 

600929 

611 

399071 

15 

40 

569172 

527 

967876 

84 

601296 

611 

398704 

14 

47 

569488 

527 

967826 

84 

601662 

611 

398338 

13 

48 

569804 

526 

967775 

84 

602029 

610 

397971 

12 

49 

570120 

526 

967725 

84 

602395 

610 

397605 

11 

50 

570435 

525 

967674 

84 

602761 

610 

397239 

10 

51 

9570751 

525 

9-967624 

84 

9603127 

609 

10396873 

9 

52 

571066 

524 

967573 

84 

603493 

609 

396507 

8 

53 

571380 

524 

967522 

85 

603858 

609 

396142 

7 

54 

571695 

523 

967471 

85 

604223 

608 

395777 

6 

55 

572009 

523 

967421 

85 

604588 

608 

395412 

5 

56 

572323 

523 

967370 

85 

604953 

607 

395047 

4 

57 

572636 

522 

967319 

85 

605317 

607 

394683 

3 

58 

572950 

522 

967268 

85 

605682 

607 

394318 

2 

59 

573263 

521 

967217 

85 

606046 

606 

393954 

1 

60 

573575 

521 

967166 

85 

606410 

606 

393590 

0 

(22  Degrees.)     A    TABLE    OF    LOGARITHMIC 


M. 

Sine 

D. 

Cosine 

1  D- 

Tan?. 

a 

Cotan<r. 

■~1 

0 

9573575 

521 

9-967166 

85 

9-606410 

606 

10393590 

60  ) 

1 

573888 

520 

967115 

85 

606773 

606 

393227 

59 

2 

574200 

520 

967064 

85 

607137 

605 

392863 

58  ) 

3 

574512 

519 

967013 

85 

607500 

605 

392500 

57, 
56  > 
55  ) 

4 

574824 

519 

966961 

85 

607863 

604 

392137 

5 

575136 

519 

966910 

85 

608225 

604 

391775 

6 

575447 

518 

966859 

85 

608588 

604 

391412 

54  ) 

7 

575758 

518 

966808 

85 

608950 

603 

391050 

53  ) 

8 

576069 

517 

966756 

86 

609312 

603 

390688 

52  > 

9 

576379 

517 

966705 

86 

609674 

603 

390326 

51  > 

10 

576689 

516 

966053 

86 

610036 

602 

389964 

50  ) 

>  11 

9576999 

516 

9-966602 

86 

9-610397 

602 

10-389603 

49  ) 

)  12 

577309 

516 

966550 

86 

610759 

602 

389241 

48  ) 

13 

577618 

515 

966499 

86 

611120 

601 

388880 

47  / 

14 

577927 

515 

966447 

86 

61 1480 

601 

388520 

46  ) 

15 

578236 

514 

966395 

86 

611841 

601 

388159 

45/ 

16 

578545 

514 

966344 

86 

612201 

600 

387799 

44/ 

►  17 

578853 

513 

966292 

86 

612561 

600 

387439 

43  > 

|  18 

579162 

513 

966240 

86 

612921 

600 

387079 

42  > 

19 

579470 

513 

966188 

86 

613281 

599 

386719 

41  > 

>  20 

579777 

512 

966136 

86 

613641 

599 

386359 

40  > 

>  21 

9-580085 

512 

9  966085 

87 

9-614000 

598 

10-386000 

39  ( 

22 

580392 

511 

966033 

87 

614359 

598 

385641 

38  ( 

>  23 

580699 

511 

965981 

87 

614718 

598 

385282 

37  ( 

24 

581005 

511 

965928 

87 

615077 

597 

384923 

36  ( 

25 

581312  . 

r  510 

965876 

87 

615435 

597 

384565 

35  ( 

26 

581618 

510 

965824 

87 

615793 

597 

384207 

34  I 

27 

581924 

509 

965772 

87 

616151 

596 

383849 

33  ( 

28 

582229 

509 

965720 

87 

616509 

596 

383491 

32  I 

29 

582535 

509 

965668 

87 

616867 

5% 

383133 

31  / 

30 

582840 

508 

965615 

87 

617224 

595 

382776 

30  ( 

31 

9583145 

508 

9-965563 

87 

9-617582 

595 

10-382418 

29  < 

32 

583449 

507 

965511 

87 

617939 

595 

382061 

28  S 

33 

583754 

507 

965458 

87 

618295 

594 

381705 

27  \ 

,  34 

584058 

506 

965406 

87 

618652 

594 

381348 

26  S 

35 

584361 

506 

965353 

88 

619008 

594 

380992 

25  < 

36 

584665 

506 

965301 

88 

619364 

593 

380636 

24  \ 

37 

584968 

505 

965248 

88 

619721 

593 

380279 

23  ( 

38 

585272 

505 

965195 

88 

620076 

593 

379924 

22  ( 

39 

585574 

504 

965143 

88 

620432 

592 

379568 

21  < 

! 40 

585877 

504 

965090 

88 

620787 

592 

379213 

20  ( 

,  41 

9-586179 

503 

9-965037 

88 

9-621142 

592 

10-378858 

19  j 

42 

586482 

503 

964984 

88 

621497 

591 

378503 

18  ) 

>  43 

586783 

503 

964931 

88 

621852 

591 

378148 

17  ) 

>  44 

587085 

502 

964879 

88 

622207 

590 

377793 

16  J 

>  45 

587386 

502 

964826 

88 

622561 

590 

377439 

15  ) 

46 

587688 

501 

964773 

88 

622915 

590 

377085 

14  ) 

47 

587989 

502 

964719 

88 

623269 

589 

376731 

13  ) 

48 

588289 

501 

964666 

89 

623623 

589 

376377 

12  \ 

49 

588590 

500 

964613 

89 

623976 

589 

376024 

11  > 

50 

588890 

500 

964560 

89 

624330 

588 

375670 

10  \ 

51 

9-589190 

499 

9964507 

89 

9-624083 

588 

10-375317 

9  I 

>  52 

589489 

499 

9644.54 

89 

625036 

588 

374904 

8  / 

)   53 

589789 

499 

964400 

89 

625388 

587 

374612 

7  / 

)  54 

590088 

498 

964347 

89 

625741 

587 

374259 

6  / 

)  55 

590387 

498 

964294 

89 

620093 

587 

373907 

5/ 

)  56 

590686 

497 

964240 

89 

626445 

586 

373555 

4  I 

)  57 

590984 

497 

964187 

89 

626797 

586 

3732(13 

3/ 

)  58 

591282 

497 

964133 

89 

627149 

586 

372851 

2  ) 

>  59 

591580 

496 

964080 

89 

627501 

585 

372499 

1  I 

)   60 

591878 

496 

964026 

89 

627852 

585 

372148 

0  ) 

|       Coiine 


67  Degrees. 


sines    and   tangents.     (23  Degrees.) 


S   M.  I 


Sine 


I       1).       I       Cosine 


0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
IS 
18 
17 
18 
lit 
20 

SI 

22 
S3 
24 
SS 

S6 

27 
S8 
S9 

30 

3] 

33 
33 
34 

35 
36 

37 
38 
3iJ 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 

<  52 

;  53 

j  54 
r  55 
?  56 

<  57 
(  58 

59 
60 


9-591878 
592176 
592473 
592770 
593067 
593363 


593955 
594251 
594547 
504842 

9595137 

595432 
595727 
590021 
596315 
596009 
59(3903 
597196 
597490 
597783 

9-598075 
598368 
598660 
598952 
599244 
599536 
599827 
600118 
600409 
600700 

9-600990 
601280 
601570 
601860 
602150 
602439 
602728 
603017 
603305 
603594 


604170 
604457 
604745 
6J05032 
605319 
605606 
605892 
606179 
606465 
9-606751 
607036 
607322 
607607 
607892 
608177 
60F4H1 
608745 
609029 
609313 


495 
495 
495 
494 
494 
493 
493 
493 
492 
492 
491 
491 
491 
490 
490 
489 
489 
489 
488 
488 
487 
487 
487 
486 
486 
485 
485 
485 
484 
484 
484 
483 
483 
482 
482 
482 
481 
481 
481 
480 
480 
479 
479 
479 
478 
478 
478 
477 
477 
476 
476 
476 
475 
475 
474 
474 
474 
473 
473 
473 


9-964026 
963972 
963919 
963865 
963811 
963757 
963704 
963650 
963596 
963542 
963488 

9-963434 
963379 
963325 
963271 
903217 
903163 
963108 
963054 
9(52999 
902945 

9-962890 
902836 
902781 
962727 
962072 
962617 
962562 
963508 
962453 
962398 

9-962343 
902288 
902233 
902178 
962123 
902067 
962012 
961957 
961902 
961846 

9-961791 
961735 
961680 
961624 
961569 
961513 
961458 
961402 
961340 
961290 

9-961235 
961179 
961123 
961067 
961011 
960955 
9M899 
960843 
960786 
960730 


90 
90 
90 
90 
91 
91 
91 
91 
91 
91 
91 
91 
91 
91 
91 
91 
91 
92 
92 
92 
92 
92 
92 
92 
92 
92 
92 
92 
92 
92 
92 
93 
93 
93 
93 
93 
93 
93 
93 
93 
93 
93 
93 
93 
93 
94 
94 


Tan?. 

D.   | 

9-627852  1 

585 

628203 

585 

628554 

585 

628905 

584 

629255 

584 

629606 

583 

629956 

583 

630306 

583 

630656 

583 

631O05 

582 

631355 

582 

9-631704 

582 

632053 

581 

632401 

581 

632750 

581 

633098 

580 

633447 

580 

633795 

580 

634143 

579 

634490 

579 

634838 

579 

9-635185 

578 

635532 

578 

635879 

578 

636226 

577 

636572 

577 

636919 

577 

637265 

577 

637611 

576 

637956 

576 

638302 

576 

9-638647 

575 

638992 

575 

639337 

575 

639682 

574 

6401,27 

574 

640371 

574 

640716 

573 

641060 

573 

641404 

573 

641747 

572 

9-642091 

572 

642434 

572 

642777 

572 

643120 

571 

643463 

571 

643806 

571 

644148 

570 

644490 

570 

-644832 

570. 

645174 

569 

9-645516 

569 

645857 

569 

646199 

569 

646540 

508 

64H881 

568 

647222 

568 

647562 

567 

647903 

567 

648243 

567 

648583 

566 

10-368296 
307947 
307599 
367250 
366902 
366553 
366205 
365857 
365510 
365162 

10-364815 
364468 
364121 
363774 
363428 
363081 
362735 
302389 
362044 
361698 


66  Degrees. 


160 


(24  Degrees.)      A    TABLE     OP    LOGARITHMIC 


JML 

Sine 

D. 

Cosine 

D.   , 

Tansr. 

D. 

Cotang. 

"H 

/  o 

9-609313 

473 

9-960730 

94 

9-648583 

566 

10-351417 

60) 

)  1 

609597 

472 

960674 

94 

648923 

566 

351077 

59 

2 

609880 

472 

960618 

94 

649263 

566 

350737 

58  ; 

57 

56) 

l  3 

610164 

472 

960561 

94 

649602 

566 

350398 

)  4 

610447 

471 

960505 

94 

649942 

565 

350058 

)    5 

610729 

471 

960448 

94 

65(1281 

565 

349719 

55  S 

>  6 

611012 

470 

960392 

94 

650620 

565 

349380 

54  ) 

7 

611294 

470 

960335 

91 

650959 

564 

349041 

53  S 

)  8 

611576 

470 

960279 

94 

651297 

564 

348703 

52 
51  ( 

9 

611858 

469 

960222 

94 

651636 

564 

348364 

>10 

612140 

469 

960165 

94 

651974 

563 

348026 

50  ( 

;  n 

9-612421 

469 

9960109 

95 

9-652312 

563 

10-347688 

49  j 

)  12 

612702 

468 

960052 

95 

652650 

563 

347350 

48) 

)  13 

612983 

468 

959995 

95 

652988 

563 

347012 

47) 

/  14 

613264 

467 

959938 

95 

653326 

562 

346674 

46) 

15 

613545 

467 

959882 

95 

653663 

562 

346337 

45) 

,  16 

613825 

467 

959825 

95 

654000 

562 

346000 

44) 

)  17 

614105 

466 

959768 

95 

654337 

561 

345663 

43) 

)  18 

614385 

466 

959711 

95 

654674 

561 

345326 

42) 

}  19 

614665 

466 

959654 

95 

655011 

561 

344989 

41) 

)   20 

614944 

465 

959596 

95 

655348 

561 

344652 

40  J 

! 21 

9-615223 

465 

9959539 

95 

9655684 

560 

10-344316 

39  j 

?  22 

615502 

465 

959482 

95 

656020 

560 

343980 

38) 

/  23 

615781 

464 

959425 

95 

656356 

560 

343644 

37) 

I   24 

616060 

464 

959368 

95 

656692 

559 

343308 

36) 

)  25 

616338 

464 

959310 

96 

657028 

559 

342972 

35) 

)  26 

616616 

463 

959253 

96 

657364 

559 

342636 

34) 

)  27 

616894 

463 

959195 

96 

657699 

559 

342301 

33) 

)  28 

617172 

462 

959138 

96 

658034 

558 

341966 

32) 

)  29 

617450 

462 

959081 

96 

658369 

558 

341631 

31) 

)  30 

617727 

462 

959023 

96 

658704 

558 

341296 

30  J 

?  31 

9-618004 

461 

9-958965 

96 

9-659039 

558 

10-340961 

29) 

(   32 

618281 

461 

958908 

96 

659373 

557 

340627 

28) 

?  33 

618558 

461 

958850 

96 

659708 

557 

340292 

27) 

(  34 

618834 

460 

958792 

96 

660042 

557 

339958 

26) 

)   35 

619110 

460 

958734 

96 

660376 

557 

339624 

25) 

I   36 
I   37 

619386 

460 

958677 

96 

660710 

556 

339290 

24) 

619662 

459 

958619 

96 

661043 

556 

338957 

23) 

)  38 

619938 

459 

958561 

96 

661377 

556 

338623 

22) 

)  39 

620213 

459 

958503 

97 

661710 

555 

338290 

21) 

)  40 

620488 

458 

958445 

97 

662043 

555 

337957 

20) 

S  41 

9620763 

458 

9-958387 

97 

9662376 

555 

10-337024 

19? 

(  42 

621038 

457 

958329 

97 

662709 

554 

337291 

18) 

(  43 

621313 

457 

958271 

97 

663042 

554 

336958 

17) 

(   44 

621587 

457 

958213 

97 

663375 

554 

336625 

16) 

(   45 

621861 

456 

958154 

97 

663707 

554 

336293 

15  J 

<  46 

622135 

456 

958096 

97 

664039 

553 

335961 

14  ) 

(   47 

622409 

456 

958038 

97 

664371 

553 

335629 

13  ) 

/  48 

622682 

455 

957979 

97 

664703 

553 

335297 

12) 

I   49 

622956 

455 

957951 

97 

665035 

553 

334965 

11  ) 

j  50 

623229 

455 

957863 

97 

665366 

552 

334634 

10) 

i  51 

9-623502 

454 

9-957804 

97 

9-665697 

652 

10-334303 

9( 

<  52 

623774 

454 

957746 

98 

666029 

552 

3331>71 

8( 

<  53 

624047 

454 

957687 

98 

666360 

551 

333640 

7) 

<  54 

624319 

453 

957628 

98 

666691 

551 

333309 

6) 

(  55 

624591 

453 

957570 

98 

667021 

551 

332979 

5) 

(  56 

624863 

453 

957511 

98 

667352 

551 

332648 

4 

<  57 

625135 

452 

957452 

98 

667683 

550 

332318 

3) 

(  58 

625406 

452 

957393 

98 

668013 

550 

331987 

2  J 

(  59 

625677 

452 

957335 

98 

668343 

550 

331657 

1  ) 

(   60 

625948 

451 

957276 

98 

668672 

550 

331328 

1 

|   Cosine 



Sine 

^~- 

Cotang. 

' 

i    Tang. 

65  Degrees. 


SINES    AND    TANGENTS.     (25  Degrees.) 


1G1 


Sine 

D. 

Cosine 

■_~dT" 

Tang. 

D. 

Cotang. 

J 

9-625948 

451 

9-957276 

98 

9-668673 

550 

10-331327 

60  ( 

620219 

451 

957217 

98 

669002 

549 

330998 

59  ( 

626490 

451 

957158 

98 

669332 

549 

330668 

58  ( 

626760 

450 

957099 

98 

669661 

549 

330339 

57  ( 

627030 

450 

957040 

98 

669991 

548 

330009 

56  ( 

627300 

450 

956981 

98 

670320 

548 

329680 

55  ( 

627570 

449 

956921 

99 

670649 

548 

329351 

54  ( 

627840 

449 

956862 

99 

670977 

548 

329023 

53  ( 

628109 

449 

956803 

99 

671306 

547 

328694 

52  ( 

628378 

448 

956744 

99 

671634 

547 

328366 

51  ( 

628647 

448 

956684 

99 

671963 

547 

328037 

50  , 

9-628916 

447 

9956625 

99 

9-672291 

547 

10-327709 

49  { 

629185 

447 

956566 

99 

672619 

546 

327381 

48  ) 

629453 

447 

956506 

99 

672947 

546 

327053 

47  ( 

629721 

446 

956447 

99 

673274 

546 

388726 

46  < 

629989 

446 

956387 

99 

673602 

546 

326398 

45  ( 

630257 

446 

956327 

99 

673929 

545 

326071 

44  ( 

630524 

446 

956208 

99 

674257 

545 

325743 

43  ( 

630792 

445 

956208 

100 

674584 

545 

325416 

42  ( 

631059 

445 

9.56148 

100 

674910 

544 

325090 

41  ( 

631326 

445 

956089 

100 

675237 

544 

324763 

40  j 

9*631593 

444 

9-956029 

100 

9-675564 

544 

10324436 

39  ) 

631859 

444 

955969 

100 

675890 

544 

324110 

38  ) 

632125 

444 

955909  ► 

100 

676216 

543 

323784 

37  ) 

632392 

443 

955849 

100 

676543 

543 

323457 

36  ) 

632658 

443 

955789 

100 

676809 

543 

323131 

35  ) 

632923 

443 

955729 

100 

677194 

543 

322806 

34  ) 

633189 

442 

955669 

100 

677520 

542 

322480 

33  ) 

633454 

442 

955009 

100 

677846 

542 

322154 

32  \ 

633719 

442 

955548 

100 

678171 

542 

321829 

31  ) 

633984 

441 

955488 

100 

678496 

542 

321504 

30  S 

9'634249 

441 

9955428 

101 

9*678821 

541 

10-321179 

29  ) 

634514 

440 

955308 

101 

679146 

541 

320854 

28  ) 

634778 

440 

955307 

101 

679471 

541 

320529 

27  ) 

635042 

440 

955247 

101 

679795 

541 

320205 

26  ) 

635306 

439 

955186 

101 

680120 

540 

319880 

25  / 

635570 

439 

955126 

101 

680444 

540 

319556 

24  ) 

635834 

439 

955005 

101 

680768 

540 

319232 

23 

636097 

438 

955005 

101 

681092 

540 

318908 

22  ) 

636360 

438 

954944 

101 

681416 

539 

318584 

21  ) 

636623 

438 

954883 

101 

681740 

539 

318260 

20 

9'636886 

437 

9-954823 

101 

9682063 

539 

10-317937 

19  ) 

637148 

437 

954702 

101 

682387 

539 

317613 

18  ) 

637411 

437 

954701 

101 

682710 

538 

317290 

17 

16  ) 

637673 

437 

954040 

101 

683033 

538 

310907 

637935 

436 

954579 

101 

683356 

538 

310644 

15 

638197 

436 

954518 

102 

683679 

538 

316321 

14  ) 

6384.58 

436 

954457 

102 

684001 

537 

315999 

13  ) 

638720 

435 

954396 

102 

684324 

537 

315676 

12  ) 

638981 

435 

954335 

102 

684646 

537 

315354 

11  / 

639242 

435 

954274 

102 

684968 

537 

315032 

io ) 

9-639503 

434 

9-954213 

102 

9-685290 

536 

10-314710 

9\ 

639764 

434 

954152 

102 

685612 

536 

314388 

8 

640024 

434 

954090 

102 

685934 

536 

314066 

7  ( 

640284 

433 

954029 

102 

686255 

536 

313745 

G( 

640544 

433 

953968 

102 

686577 

535 

313423 

5 

640804 

433 

953906 

102 

686898 

535 

313102 

4 

641064 

432 

953845 

102 

687219 

535 

312781 

3 

641324 

432 

953783 

102 

687540 

535 

312460 

2 

641584 

432 

953722 

103 

687861 

534 

312139 

1  ( 

641842 

431 

953600 

103 

688182 

534 

311818 

0  ( 

64  Degrees. 


/x 


162 


(26  Degrees.)     A   TABLE    OF   LOGAR1TMIC 


Sine 

1   D- 

Cosine 

D. 

Tang. 

D. 

Cotang. 

9-641842 

431 

9-953660 

103 

9-688182 

534 

10'311818 

60 

642101 

431 

953599 

103 

688502 

534 

311498 

59 

642360 

421 

953537 

103 

688823 

534 

311177 

58 

642618 

430 

953475 

103 

689143 

533 

310857 

57 

642877 

430 

953413 

103 

689403 

533 

310537 

56 

643135 

430 

953352 

103 

689783 

533 

310217 

55 

643393 

430 

953290 

103 

690103 

533 

309897 

54 

643650 

429 

953228 

103 

690423 

533 

309577 

53 

643908 

429 

953166 

103 

690742 

532 

309258 

52 

644165 

429 

953104 

103 

691062 

532 

308938 

51 

644423 

428 

953042 

103 

691381 

532 

308619 

50 

9644680 

428 

9-952980 

104 

9-691700 

531 

10-308300 

49 

644936 

428 

952918 

104 

692019 

531 

307981 

48 

645193 

427 

952855 

104 

692338 

531 

307662 

47 

645450 

427 

952793 

104 

692656 

531 

307344 

46 

645706 

427 

952731 

104 

692975 

531 

307025 

45 

045962 

426 

952669 

104 

693293 

530 

306707 

44 

646218 

420 

952606 

104 

693612 

530 

306388 

43 

646474 

426 

952544 

104 

693930 

530 

306070 

42 

646729 

425 

952481 

104 

694248 

530 

305752 

41 

646984 

425 

952419 

104 

694566 

529 

305434 

40 

9-647240 

425 

9-952356 

104 

9-694883 

529 

10-305117 

38 

647494 

424 

952294 

104 

695201 

529 

304799 

38 

647749 

424 

952231 

104 

695518 

529 

304482 

37 

648004 

424 

952168 

105 

695836 

529 

304164 

36 

648258 

424 

952106 

105 

696153 

528 

303847 

35 

648512 

423 

952043 

105 

696470 

528 

303530 

34 

648766 

423 

951980 

105 

696787 

528 

303213 

33 

649G20 

423 

951917 

105 

697103 

528 

302897 

32 

649274 

422 

951854 

105 

697420 

527 

302580 

31 

649527 

422 

951791 

105 

697736 

527 

302264 

30 

9649781 

422 

9-951728 

105 

9-698053 

527 

10-301947 

2!) 

650034 

422 

951665 

105 

6983C9 

527 

301631 

28 

650287 

421 

951602 

105 

698C85 

526 

301315 

27 

650539 

421 

951539 

105 

699001 

526 

300999 

26 

650792 

421 

951476 

1C5 

699316 

526 

300684 

25 

651044 

420 

951412 

105 

699632 

526 

300368 

24 

651297 

420 

951349 

106 

699947 

526 

300053 

23 

651549 

420 

951286 

106 

700203 

525 

299737 

22 

651800 

419 

951222 

106 

700578 

525 

299422 

Si 

652052 

419 

951159 

106 

700893 

525 

299107 

81 

9-652304 

419 

9-951096 

1C6 

9-701208 

524 

10-298792 

18 

652555 

418 

951032 

106 

701523 

524 

298477 

18 

652806 

418 

950968 

1.06 

701837 

524 

298163 

17 

653057 

418 

950905 

106 

702152 

524 

297848 

Hi 

653308 

418  . 

950841 

K-6 

702466 

524 

297534 

15 

653558 

417 

950778 

106 

702780 

523 

297220 

14 

653808 

417. 

950714 

106 

703095 

523 

296905 

13 

654059 

417 

950650 

106 

703409 

523 

296591 

IS 

654309 

416 

950586 

106 

703723 

523 

296277 

11 

654558 

416 

950522 

107 

704036 

522 

295964 

10 

654808 

416 

9-950458 

107 

9704350 

522 

10-295650 

9 

655058 

416 

950394 

107 

704663 

522 

295337 

8 

655307 

415 

950330 

107 

704977 

522 

295023 

7 

655556 

415 

950266 

107 

705290 

522 

294710 

6 

655805 

415 

950202 

107 

705603 

521 

294397 

5 

656054 

414 

950138 

107 

705916 

521 

294084 

4 

656302 

414 

950074 

107 

706228 

521 

293772 

3 

65655] 

414 

950010 

107 

706541 

521 

293459 

2 

656709 

413 

949945 

107 

706854 

521 

293146 

1 

657047 

413 

949881 

107 

707166 

520 

292834 

0 

Cosine      | 


63  Degrees. 


SINES    AND    TANGENTS.      (27  Degrees.) 


1G3 


M. 

Sine 

D. 

Cosine 

D.   | 

Tang-. 

D. 

Cotangv 

0 

9657047 

413 

9-949881 

107 

9-707166 

520 

10-292834 

(50 

1 

657295 

413 

949816 

107 

707478 

520 

292522 

59 

2 

657542 

412 

949752 

'107 

707790 

520 

292210 

58 

S 

657790 

412 

949688 

108 

708102 

520 

291898 

57 

4 

658037 

412 

949623 

108 

708414 

519 

291586 

98 

5 

658284 

412 

949558 

108 

708726 

519 

291274 

55 

6 

658531 

411 

949494 

108 

709037 

519 

290963 

54 

7 

658778 

411 

949429 

108 

709349 

519 

290651 

53 

8 

659025 

411 

9493(54 

108 

709660 

519 

290340 

52 

9 

659271 

410 

949300 

108 

709971 

318 

290029 

51 

10 

659517 

410 

949235 

108 

710282 

518 

289718 

50 

11 

9-659763 

410 

9-949170 

108 

9-710593 

518 

10-289407 

49 

12 

660009 

409 

949105 

108 

710904 

518 

289096 

48 

13 

660255 

409 

949040 

108 

711215 

518 

288785 

47 

14 

660501 

409 

948975 

108 

711525 

517 

288475 

4(5 

15 

6(50746 

409 

948910 

108 

711836 

517 

288164 

45 

16 

660991 

408 

948845 

108 

712146 

517 

287854 

44 

17 

661236 

408 

948780 

109 

712456 

517 

287544 

1:1 

13 

661481 

408 

948715 

109 

712766 

516 

287234 

1-2 

19 

661726 

407 

948650 

109 

713076 

516 

286924 

41 

20 

661970 

407 

948584 

109 

713386 

516 

286614 

40 

21 

9662214 

407 

9-948519 

109 

9-713696 

516 

10-286304 

39 

22 

662459 

407 

948454 
948388~ 

109 

714005 

516 

285995 

39 

23 

662703 

406 

109 

714314 

515 

285686 

37 

24 

662946 

406 

948323 

109 

714624 

515 

285376 

3(5 

25 

663190 

406 

948257 

109 

714933 

515 

285067 

35 

20 

663433 

405 

948192 

109 

715242 

515 

284758 

34 

27 

663677 

405 

948126 

109 

715551 

514 

284449 

33 

28 

663920 

405 

948060 

109 

715860 

514 

284140 

3-2 

29 

664162 

405 

947995 

110 

716168 

514 

283832 

31 

30 

664406 

404 

947929 

110 

716477 

514 

283523 

3Q 

31 

9664648 

404 

9-947863 

110 

9-716785 

514 

10-283215 

29 

32 

664891 

404 

947797 

110 

717093 

513 

282907 

28 

33 

665133 

403 

947731 

110 

717401 

513 

282599 

•27 

34 

665375 

403 

947665 

110 

717709 

513 

282291  • 

95 

35 

665617 

403 

947600 

110 

718017 

513 

281983 

35 

30 

665859 

402 

917533 

110 

718325 

513 

281(575 

24 

37 

666100 

402 

947467 

110 

718633 

512 

281367 

■Zi 

38 

666342 

402 

947401 

110 

718940 

512 

281060 

22 

39 

666583 

402 

947335 

110 

719248 

512 

280752 

SI 

40 

666824 

401 

947269 

110 

719555 

512 

280445 

80 

41 

9  667065 

401 

9-947203 

110 

8-719862 

512 

10-280138 

19 

42 

667305 

401 

847136 

111 

720169 

511 

279831 

18 

43 

667546 

401 

947070 

111 

720476 

511 

279524 

17 

44 

667786 

400 

947004 

111 

720783 

511 

279217 

1(5 

45 

668027 

400 

940937 

111 

721089 

511 

278911 

15 

46 

668267 

400 

946871 

111 

721396 

511 

278604 

14 

47 

668506 

309 

940804 

111 

721702 

510 

278298 

13 

48 

668746 

309 

946738 

111 

722009 

510 

277991 

1-2 

49 

668986 

399 

946671 

111 

722315 

510 

277685 

11 

50 

669225 

399 

946604 

111 

722621 

510 

277379 

10 

51 

9-669464 

398 

9-946538 

111 

9-722927 

510 

10-277073 

9 

52 

669703 

398 

946471 

111 

723232 

509 

276768 

8 

53 

669942 

398 

946404 

111 

723538 

509 

276462 

7 

54 

670181 

397 

946337 

111 

723844 

509 

276156 

6 

55 

670419 

397 

946270 

112 

724149 

509 

275851 

5 

56 

6706.58 

397 

946203 

112 

724454 

509 

275546 

4 

57 

670896 

397 

946136 

112 

724759 

508 

275241 

3 

58 

671134 

396 

946069 

112 

725065 

508 

274935 

2 

59 

671372 

396 

940002 

112 

725369 

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274631 

1 

60 

671609 

396 

945935 

112 

725674 

508 

274326 

0 

r 


I       Cosine       | 


Sine        | 


|       Cotang. 


62  Degrees. 


164 


(28  Degrees.)     a    i'able    of    logarithmic 


Cm. 

Sine 

D. 

Cosine 

D. 

Tang. 

D. 

Cotan^. 

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9*671609 

396 

9-945935 

112 

9-725674 

508 

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395 

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112 

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508 

274021 

59  S 

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395 

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112 

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507 

273716 

58  ) 

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395 

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112 

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672558 

395 

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112 

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507 

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672795 

394 

945598 

112 

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507 

272803 

55) 

?  6 

673032 

394 

945531 

112 

727501 

507 

272499 

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)  7 

673268 

394 

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113 

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506 

272195 

53  ) 

J  8 

673505 

394 

945396 

113 

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506 

271891 

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)  9 

673741 

393 

945328 

113 

728412 

506 

271588 

51  ) 

i  10 

673977 

393 

945261 

113 

728716 

506 

271284 

50  S 

1  U 

9674213 

393 

9-945193 

113 

9-729020 

506 

10-270980 

49) 

/  12 

674448 

392 

945125 

113 

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505 

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674684 

392 

945058 

113 

729626 

505 

270374 

47  I 

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674919 

392 

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113 

729929 

505 

270071 

46/ 

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675155 

392 

944922 

113 

730233 

505 

269767 

45/ 

)  16 

675390 

391 

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113 

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505 

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675624 

391 

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113 

730838 

504 

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391 

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113 

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391 

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504 

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503 

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/  26 

677731 

389 

944172 

114 

733558 

503 

266442 

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/  27 

677964 

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944104 

114 

733860 

502 

266140 

33  ( 

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114 

734162 

502 

265838 

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678430 

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386 

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<  38 

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500 

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385 

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115 

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500 

262529 

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680982 

385 

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115 

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500 

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9681213 

385 

9-943141 

115 

9-738071 

500 

10-261929 

19/ 

i  42 

681443 

384 

943072 

115 

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500 

261629 

18/ 

5  43 

681674 

384 

943003 

115 

738671 

499 

261329 

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<  44 

681905 

384 

942934 

115 

738971 

409 

261029 

16  / 

<  45 

682135 

384 

942864 

115 

739271 

499 

260729 

15/ 

(  46 

682365 

383 

942795 

116 

739570 

499 

260430 

14/ 

<  47 

682595 

383 

942726 

116 

739870 

499 

260130 

13/ 

<  48 

682825 

383 

942656 

116 

740169 

499 

259831 

12/ 

(  49 

683055 

383 

942587 

116 

740468 

498 

259532 

11  ( 

j  50 

683284 

382 

942517 

116 

740767 

498 

259233 

10/ 

)  51 

9683514 

382 

9-942448 

116 

9-741066 

498 

10-258934 

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683743 

382 

942378 

116 

741365 

498 

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53 
J  54 

683972 

382 

942308 

116 

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498 

258336 

7{ 

684201 

381 

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116 

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497 

258038 

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684430 

381 

942169 

116 

742261 

497 

257739 

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684658 

381 

942099 

116 

742559 

497 

257441 

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380 

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116 

742858 

497 

257142 

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S  58 

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380 

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116 

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380 

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117 

743454 

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256546 

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685571 

380 

941819 

117 

743752 

496 

256248 

0  J 

{ 

Cosine 

Sine 

|   Cotang. 

}^^~> 

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XT) 

61  Degrees. 


BINES   AND   TANGENTS.     (29  Degrees.) 


165 


0 

9685571 

380 

9-941819 

117 

9-743752 

496 

10-256248 

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1 

685799 

379 

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117 

744050 

496 

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2 

686027 

379 

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117 

744348 

496 

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58) 

3 

686254 

379 

941609 

117 

744645 

496 

255355 

57) 

4 

686482 

379 

941539 

117 

744943 

496 

255057 

56) 

)  5 

686709 

378 

941469 

117 

745240 

496 

254760 

55) 

>   6 

686936 

378 

941398 

117 

745538 

495 

254462 

54) 

)   ' 

687163 

378 

941328 

117 

745835 

495 

254165 

53) 

8 

687389 

378 

941258 

117 

746132 

495 

253868 

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)   9 

687616 

377 

941187 

117 

746429 

495 

253571 

51) 

>10 

687843 

377 

941117 

117 

746726 

495 

253274 

50) 

Ui 

9-688069 

377 

9-941046 

118 

9747023 

494 

10-252977 

49  ( 

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688295 

377 

940975 

118 

747319 

494 

252681 

48? 

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688521 

376 

940905 

118 

747616 

494 

252384 

47? 

14 

688747 

376 

940834 

118 

747913 

494 

252087 

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688972 

376 

940763 

118 

748209 

494 

251791 

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>16 

689198 

376 

940093 

118 

748505 

493 

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689423 

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940622 

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375 

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118 

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118 

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493 

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41  ? 

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118 

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493 

250311 

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21 

9-690323 

374 

9-940338 

118 

9-749985 

493 

10-250015 

39 

22 

690548 

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118 

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374 

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119 

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373 

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26 

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373 

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119 

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248538 

34  ( 

27 

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373 

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119 

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248243 

33  ( 

28 

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752052 

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372 

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119 

752347 

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247653 

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119 

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247358 

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31 

9-692562 

372 

9-939625 

119 

9-752937 

491 

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29 

32 

692785 

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246769 

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119 

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490 

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36 

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120 

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490 

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37 

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370 

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120 

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490 

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38 

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370 

939123 

120 

754997 

490 

245003 

22  ) 

39 

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370 

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120 

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490 

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21  ) 

40 

694564 

369 

938980 

120 

755585 

489 

244415 

20 ; 

41 

9-694786 

369 

9-938908 

120 

9-755878 

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10-244122 

19 

42 

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369 

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120 

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243828 

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43 

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369 

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120 

756465 

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243535 

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695450 

368 

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120 

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368 

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120 

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489 

242948 

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368 

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120 

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488 

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368 

938475 

120 

757638 

488 

242362 

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48 

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367 

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121 

757931 

488 

242069 

12  ) 

49 

696554 

367 

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121 

758224 

488 

241776 

11  ( 

50 

696775 

367 

938258 

121 

758517 

488 

241483 

10  ) 

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9-696995 

367 

9-938185 

121 

9-758810 

488 

10-241190 

9 

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697215 

366 

938113 

121 

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487 

240898 

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697435 

366 

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121 

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487 

240605 

7\ 

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366 

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121 

759687 

487 

240313 

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j  55 

697874 

366 

937895 

121 

759979 

487 

240021 

5  ( 

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698094 

365 

937822 

121 

760272 

487 

239728 

4  < 

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698313 

365 

937749 

121 

760564 

487 

239436 

3  ( 

)58 

698532 

365 

937676 

121 

760856 

486 

239144 

2  ( 

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698751 

365 

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121 

761148 

486 

238852 

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698970 

364 

937531 

121 

761439 

486 

238561 

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Cosine 

Sine 

Cotang. 



^Tang^l 

JL) 

C 



0  Degre 

BS. 

166 


(30  Degrees.)      A  TABLE    OP   LOGARITHMIC 


(m. 

1    Sine 

1   D. 

Cosine 

1   D. 

1   Tang. 

1   D. 

Cotang. 

1     I 

1  ° 

9-698970 

364 

9937531 

121 

9-761439 

486 

10-238561 

60  ) 

>  1 

699189 

364 

937458 

122 

761731. 

486 

238269 

59  ) 

?  2 

699407 

364 

937385 

122 

762023 

486 

237977 

58  ; 

?  3 

699626 

364 

937312 

122 

762314 

486 

237686 

57  ) 

f    4 

699844 

363 

937238 

122 

762606 

485 

237394 

56  ) 

)  5 

700062 

363 

937165 

122 

762897 

485 

237103 

55  ) 

)    6 

700280 

363 

937092 

122 

763188 

485 

236812 

54  > 

)    7 

700498 

363 

937019 

122 

763479 

485 

236521 

53  ) 

)    8 

700716 

363 

936946 

122 

763770 

485 

236230 

o2  ) 

?  9 

700933 

362 

936872 

122 

764061 

485 

235939 

51  \ 

s 10 

701151 

362 

936799 

122 

764352 

484 

235648 

50  j 

\   11 

9701368 

362 

9-936725 

122 

9764643 

484 

10-235357 

49  / 

<  12 

701585 

362 

936652 

123 

764933 

484 

235067 

48  ) 

<  13 

701802 

361 

936578 

123 

765224 

484 

234776 

47  ) 

<  14 

702019 

361 

936505 

123 

765514 

484 

234486 

46  ) 

s  15 

702236 

361 

936431 

123 

765805 

484 

234195 

45  I 

(  16 

702452 

361 

936357 

123 

766095 

484 

233905 

44  / 

S  17 

702669 

360 

936284 

123 

766385 

483 

233615 

43  ) 

S  18 

702885 

360 

936210 

123 

766675 

483 

233325 

42  ( 

(  19 

703101 

360 

936136 

123 

766965 

483 

233035 

41  / 

J  20 

703317 

360 

936062 

123 

767255 

483 

232745 

40  ) 

S  21 

9-703533 

359 

9-935988 

123 

9-767545 

483 

10-232455 

39  ( 

S  22 

703749 

359 

935914 

123 

767834 

483 

232166 

38  ( 

S  23 

703964 

359 

935840 

123 

768124' 

482 

231876 

37  ( 

>  24 

734179 

359 

935766 

124 

768413 

482 

231587 

36  < 

s  25 

704395 

359 

935692 

124 

768703 

482 

231297 

35  ( 

)  26 

704610 

358 

935618 

124 

768992 

482 

231008 

34  < 

S  27 

704825 

358 

935543 

124 

769281 

482 

230719 

33  < 

S  28 

705040 

358 

935469 

124 

769570 

482 

230430 

32  ( 

S  29 

705254 

358 

935395 

124 

769860 

481 

230140 

31  ( 

J  30 

705469 

357 

935320 

124 

770148 

481 

229852 

30  ( 

?  31 

9-705683 

357 

9935246 

124 

9770437 

481 

10-229563 

29  ) 

>  32 

705898 

357 

935171 

124 

770726 

481 

229274 

28  ) 

>  33 

706112 

357 

935097 

124 

771015 

481 

228985 

27 

)  34 

706326 

356 

935022 

124 

771303 

481 

228697 

26  ) 

)  35 

706539 

356 

934948 

124 

771592 

481 

228408 

25 
24  1 
23 

22  ) 

)   36 

706753 

356 

934873 

124 

771880 

480 

228120 

I  37 

706967 

356 

934798 

125 

772168 

480 

227832 

)  38 

707180 

355 

934723 

125 

772457 

480 

227543 

>  39 

707393 

355 

934649 

125 

772745 

480 

227255 

21  ) 

>  40 

707606 

355 

934574 

125 

773033 

480 

226967 

20  ) 

(  41 

9707819 

355 

9934499 

125 

9773321 

480 

10-226679 

19  ( 

<  42 

708032 

354 

934424 

125 

773608 

479 

226392 

18  / 

S  43 

708245 

354 

934349 

125 

773896 

479 

226104 

17  ( 

(  44 

708458 

354 

934274 

125 

774184 

479 

225816 

16  / 

(  45 

708670 

354 

934199 

125 

774471 

479 

225529 

15  / 

(   46 

708882 

353 

934123 

125 

774759 

479 

225241 

14  ( 

S  47 

709094 

353 

934048 

125 

775046 

479 

224954 

13  I 

I   48 

709306 

353 

933973 

125 

775333 

479 

224667 

12  ? 

(  49 

709518 

353 

933898 

126 

775621 

478 

221379 

11  \ 

J  50 

709730 

353 

933822 

126 

775908 

478 

224092 

10 

)   51 

9-709941 

352 

9-933747 

126 

9-776195 

478 

10223805 

9  ! 

S  52 

710153 

352 

933671 

126 

776482 

478 

223518 

8  ) 

)  53 
)   54 

710364 

352 

933596 

126 

776769 

478 

223231 

7  { 

710575 

352 

933520 

126 

777055 

478 

222945 

6  S 

S  55 

710786 

351 

933445 

126 

777342 

478 

222658 

5  S 

\   56 

710997 

351 

933369 

126 

777628 

477 

222372 

4  s 

)  57 

711208 

351 

933293 

126 

777915 

477 

222085 

3  S 

S  58 

711419 

351 

933217 

126 

778201 

477 

221799 

2  S 

S  59 

711629 

350 

933141 

126 

778487 

477 

221512 

1  < 

60 

711839 

350 

933066 

126 

778774 

477 

221226 

0  ( 

L~ 

Cosine 

~— -, 

Sine 

Cotang.  | 

Tang. 

^MJ 

Degrees. 


SINES    an/)    TiNuENTS.    (31  Degrees.) 


167 


CmT 

Sine 

D. 

Cosine 

D.   | 

T.  ng.   | 

D.   | 

Cotang. 

f   ° 

9-711839 

350 

9-933066 

126 

9-778774 

477 

1U-221226 

00  } 

I   1 

712050 

350 

932990 

127 

779000 

477 

220940 

60 

>  2 

712200 

350 

932914 

127 

779346 

476 

220054 

Rj) 

)  3 

712409 

349 

9328:18 

127 

779032 

476 

220368 

57) 

)  4 

712079 

349 

932702 

127 

779918 

476 

220082 

5'J 

)  5 

712889 

349 

932085 

127 

780203 

476 

219797 

55/ 

>  6 

713098 

349 

932809 

127 

780489 

476 

219511 

54^ 

)  7 

'  713303 

349 

932533 

127 

78J775 

470 

219225 

53< 

)  8 

713517 

348 

932457 

127 

781000 

470 

213940 

52? 

>  9 

713726 

343 

932380 

127 

781346 

475 

218654 

51 ; 

(  I0 

713935 

348 

9323)4 

127 

781631 

475 

218369 

50/ 

)  11 

9714144 

343 

9-932228 

127 

9-781916 

475 

10-218084 

49  I 

(  12 

714352 

347 

932151 

127 

7822U1 

475 

217799 

48  ( 

(  13 

714561 

347 

932075 

128 

782486 

475 

217514 

47  ( 

(  14 

714769 

347 

931938 

128 

782771 

475 

217229 

40  < 

(  15 

714978 

347 

931921 

128 

783056 

475 

216944 

43  ( 

(  16 

715186 

347 

931345 

128 

783341 

475 

210659 

41  ( 

(  17 

715394 

340 

931768 

128 

783026 

474 

21 6374 

43  ( 

(  18 

715602 

346 

931691 

128 

783910 

474 

216090 

42  ( 

/  19 

715809 

346 

931014 

128 

784195 

474 

215805 

41  ( 

>20 

716017 

346 

931537 

128 

784479 

474 

215521 

40  ( 

<  21 

9-718224 

345 

9-931460 

128 

9-784704 

474 

10-215230 

39  > 

(22 

716432 

345 

931383 

128 

785048 

474 

214952 

38  ) 

(23 

716039 

345 

931306 

128 

785332 

473 

214668 

37  S 

(24 

716846 

345 

931229 

129 

785010 

473 

214384 

30  ) 

(25 

717053 

345 

931152 

129 

785900 

473 

214100 

35  ( 

(26 

717259 

344 

931075 

129 

78G184 

473 

213816 

34  S 

(27 

717466 

344 

930998 

129 

780408 

473 

213532 

33  \ 

(28 

717673 

344 

930921 

129 

78G752 

473 

213248 

32  S 

(29 

717879 

344 

930843 

129 

787036 

473 

212964 

31  S 

(30 

7180a1* 

343 

930760 

129 

787319 

472 

212081 

30  \ 

31 

9718291 

343 

9-930083 

129 

9-787603 

472 

10-212397 

29) 

,32 

718497 

343 

930611 

129 

787886 

472 

212114 

28) 

S  33 

718703 

343 

933533 

129 

788170  ' 

472 

211830 

27) 

34 

718909 

343 

930450 

129 

788453 

472 

211547 

20) 

(  35 

719114 

342 

930378 

129 

788730 

472 

211264 

25) 

(30 

719320 

342 

930300 

130 

789019 

472 

210981 

24  ) 

(37 

719525 

342 

930223 

130 

789302 

471 

210098 

23) 

(38 

719730 

342 

930145 

130 

789585 

471 

210415 

22) 

(39 

719935 

341 

930007 

130 

789868 

471 

210132 

21  ) 

(40 

720140 

341 

929989 

130 

790151 

471 

209849 

20) 

>41 

9-720345 

341 

9-929911 

130 

9-790433 

471 

10-209567 

19? 

)42 

720549 

341 

929833 

130 

790716 

471 

209284 

18/ 

)43 

720754 

340 

929755 

130 

790999 

471 

209001 

17/ 

)44 

720958 

340 

929677 

130 

791281 

471 

208719 

16/ 

)45 

721162 

340 

929599 

130 

7915G3 

470 

208437 

15/ 

)  40 

721366 

340 

929521 

130 

791846 

470 

208154 

14/ 

47 

721570 

340 

929442 

130 

792128 

470 

207872 

13/ 

)  48 

721774 

339 

929364 

131 

792410 

470 

207590 

12  > 

)  49 

721978 

339 

929286 

131 

792092 

470 

207308 

11  ( 

[50 

722181 

339 

929207 

131 

792974 

470 

207026 

10) 

}51 

9-722385 

339 

9-929129 

131 

9-793250 

470 

10-200744 

9S 

)  52 

722588 

339 

929050 

131 

793538 

409 

206462 

8\ 

)  53 

722791 

338 

928972 

131 

793819 

469 

200181 

7S 

)  54 

722994 

338 

928893 

131 

794101 

469 

205899 

6  ( 

>55 

723197 

338 

928815 

131 

794383 

409 

205017 

5  ( 

)  56 

723400 

338 

928736 

131 

794664 

469 

205336 

4\ 

)57 

723603 

337 

928657 

131 

794945 

469 

205055 

3( 

)  58 

723805 

337 

928578 

131 

795227 

469 

204773 

2  ( 

)  59 

724007 

337 

928499 

131 

795508 

468 

204492 

1  ( 

>60 

724210 

337 

928420 

131 

795789 

468 

204211 

0  ( 

58  Debtees. 


168 


(32  Degrees.)     A    TABLE    OF     LOGARITHMIC 


[mT| 

S.ne    | 

D. 

Cosine   | 

D. 

Tan-. 

D. 

Cotang. 

i 

I  ° 

9724210 

337 

9928420 

132 

9795789 

468 

10-204211 

60  ? 

)    1 

724412 

337 

928342 

132 

796070 

468 

203930 

59  ( 

I    2 

724614 

336 

928263 

132 

796351 

468 

203649 

58  ( 

?  3 

724816 

336 

928183 

132 

796632 

468 

203368 

57  < 

)  4 

725017 

336 

928104 

132 

796913 

468 

203087 

56  ( 

)  5 

725219 

336 

928025 

132 

797194 

468 

202806 

55  / 

)  6 

725420 

335 

927946 

132 

797475 

468 

202525 

54  I 

;    7 

725622 

335 

927867 

132 

797755 

468 

202245 

53  < 

)  8 

725823 

335 

927787 

132 

798036 

467 

201964 

52  ( 

)  9 

726024 

335 

927708 

132 

798316 

467 

201684 

51  ( 

j  10 

726225 

335 

927629 

132 

798596 

467 

201404 

50  j 

(  11 

9726426 

334 

9-927549 

132 

9-798877 

467 

10201123 

49  S 

/  12 

726626 

334 

927470 

133 

799157 

467 

200843 

48  S 

)  13 

726827 

334 

927390 

133 

799437 

467 

200563 

47  < 

)  14 

727027 

334 

927310 

133 

799717 

467 

200283 

46  \ 

>  15 

727228 

334 

927231 

133 

799997 

466 

200003 

45  S 

?  16 

727423 

333 

927151 

133 

800277 

466 

199723 

44  S 

)  I7 

727628 

333 

927071 

133 

800557 

466 

199443 

43  \ 

)  i8 

727828 

333 

926991 

133 

800836 

466 

199164 

42  S 

)  19 

728027 

333 

926911 

133 

801116 

466 

198884 

41  S 

J  20 

728227 

333 

926831 

133 

801396 

466 

198604 

40  S 

(  21 

9728427 

332 

9-926751 

133 

9801675 

466 

10-198325 

39) 

/  22 

728026 

332 

926671 

133 

801955 

466 

198045 

38  > 

?  23 

728825 

332 

926591 

133 

802234 

465 

197766 

37  / 

/  24 

729024 

332 

926511 

134 

802513 

465 

197487 

36/ 

)  25 

729223 

331 

926431 

134 

802792 

465 

197208 

35  ) 

)  26 

729422 

331 

926351 

134 

803072 

465 

196928 

34  { 

)  27 

729621 

331 

926270 

134 

803351 

465 

196649 

33  ( 
32) 

)  28 

729820 

331 

926190 

134 

803630 

465 

196370 

>  29 

730018 

330 

926110 

134 

803908 

465 

196092 

31  / 

)  30 

730216 

330 

926029 

134 

804187 

465 

195813 

30 

<  31 

9  730415 

330 

9-925949 

134 

9804466 

464 

10195534 

29  { 

(  32 

730613 

330 

925868 

134 

804745 

464 

195255 

28  ( 

(   33 

730811 

330 

925788 

134 

805023 

464 

194977 

27  < 

<  34 

731009 

329 

925707 

134 

805302 

464 

194698 

26  < 

?  35 

731206 

329 

925626 

134 

805580 

464 

194420 

25  < 

?  36 

731404 

329 

925545 

135 

805859 

464 

194141 

24  < 

<  37 

731602 

329 

925465 

135 

806137 

464 

193863 

23  ( 

<  38 

731799 

329 

925384 

135 

806415 

463 

193585 

22  ( 

/  39 

731996 

328 

925303 

135 

806693 

463 

193307 

21  < 

(  40 

732193 

328 

925222 

135 

806971 

463 

193029 

20  ( 

<  41 

9732390 

328 

9-925141 

135 

9-807249 

463 

10-192751 

19  J 

<  42 

732587 

328 

925060 

135 

807527 

463 

192473 

18  > 

<  43 

732784 

328 

924979 

135 

807805 

463 

192195 

17  ) 

<  44 

732980 

327 

924897 

135 

808083 

463 

191917 

16  > 

(  45 

733177 

327 

924816 

135 

808361 

463 

191639 

15  > 

<  46 

733373 

327 

924735 

136 

808638 

462 

191362 

14  > 

<  47 

733569 

327 

924654 

136 

808916 

462 

191084 

13) 

(  48 

733765 

327 

924572 

136 

809193 

462 

190807 

12) 

(  49 

733961 

326 

924491 

136 

809471 

462 

190529 

11  I 

(  50 

734157 

326 

924409 

136 

809748 

462 

190252 

10) 

/  51 

9  734353 

326 

9-924328 

136 

9-810025 

462 

10189975 

9< 

/  52 

734549 

326 

924246 

136 

810302 

462 

189698 

8^ 

)  53 

734744 

325 

924164 

136 

810580 

462 

189420 

"M 

>54 

734939 

325 

924083 

136 

810857 

462 

189143 

6  \ 

)  55 

735135 

325 

924001 

136 

811134 

461 

188866 

5  l 

)  56 

735330 

325 

923919 

136 

811410 

461 

188590 

4  ( 

;  57 

735525 

325 

923837 

136 

811687 

461 

188313 

3  \ 

)  58 

735719 

324 

023755 

137 

811964 

461 

188036 

2  ( 

59 

735914 

324 

933673 

137 

812241 

461 

187759 

1 

)  60 

736109 

1  324 

923501 

137 

812517 

461 

187483 

0  ( 

U 

1   Cosine 

i^-~^> 

1   Km 

L^^ 

Cotang. 

^>~~ 

Tang. 

l_JLJ 

57  Degrees. 


sines    AND    tangents.     (33  Degrees.) 


169 


>  M. 

Sine 

.   D. 

Cosine 

D. 

Tang. 

1   D- 

1   Cotang.   |    \ 

1     ° 

9736109 

324 

J9-923591 

137 

9-812517 

461 

10-187482 

60  { 

(  1 

736303 

324 

923509 

137 

812794 

461 

187206 

59  ( 

58  ( 

S  2 

736498 

324 

923427 

137 

813070 

461 

186930 

\  3 

736692 

323 

923345 

137 

813347 

460 

186653 

57 

(    4 

736886 

323 

923263 

137 

813623 

460 

186377 

56? 

S  5 

737080 

323 

923181 

137 

813899 

460 

186101 

55/ 

'  6 

737274 

323 

923098 

137 

814175 

460 

185825 

54  ( 

\» 

737467 

323 

923016 

137 

814452 

460 

185548 

53  I 

737661 

322 

923983 

137 

814728 

460 

185272 

52  (' 

S  9 

737855 

322 

922851 

137 

815004 

460 

184996 

51  ( 

>  10 

738048 

322 

922768 

138 

815279 

460 

184721 

50/ 

i  1J 

9-738241 

322 

9-922686 

138 

9-815555 

459 

10184445 

49  ( 

)  12 

738434 

322 

922603 

138 

815831 

459 

184169 

48  < 

)  13 

738627 

321 

922520 

138 

816107 

459 

183893 

47  ( 

)  14 

738820 

321 

922438 

138 

816382 

459 

183618 

46  ( 

)  15 

739013 

321 

922355 

138 

81G658 

459 

183342 

45  f 

)  16 

739206 

321 

922272 

138 

816933 

459 

183067 

44  ( 

i   n 

739398 

321 

922189 

138 

817209 

459 

182791 

43  ( 

>  18 

739590 

320 

922106 

138 

817484 

459 

182516 

42( 

)  19 

739783 

320 

922023 

138 

817759 

459 

182241 

41  ( 

)  20 

739975 

320 

921940 

138 

818035 

458 

181965 

40  ( 

>  21 

9-740167 

320 

9  921857 

139 

9-818310 

458 

10-181690 

39  S 

/  22 

740359 

320 

921774 

139 

818585 

458 

181415 

38  S 

>  23 

740550 

319 

921691 

139 

818860 

458 

181140 

37) 

I  24 

740742 

319 

921607 

139 

819135 

458 

180865 

36  S 

/  25 

740934 

319 

921524 

139 

819410 

458 

180590 

35) 

I  26 

741125 

319 

921441 

139 

819684 

458 

180316 

34  ( 
33  (. 

I  27 

741316 

319 

921357 

139 

819959 

458 

180041 

I  28 

741508 

318 

921274 

139 

820234 

458 

179766 

32  S 

I  29 

741099 

318 

921190 

139 

820508 

457 

179492 

31$ 

/  30 

741889 

318 

921107 

139 

820783 

457 

179217 

30  j 

I   31 

9-742080 

318 

9-921023 

139 

9-821057 

457 

10178943 

29) 

<  32 

742271 

318 

920939 

140 

821332 

457 

178668 

28  ) 
27  S 

(  33 

7424G2 

317 

920856 

140 

821606 

457 

178394 

(  34 

742652 

317 

920772 

140 

821880 

457 

178120 

26) 
25) 

<  35 

742842 

317 

920688 

140 

822154 

457 

177846 

(  36 

743033 

317 

920604 

140 

822429 

457 

177571 

24) 

<  37 

743223 

317 

920520 

140 

822703 

457 

177297 

23) 

(  38 

743413 

316 

920436 

140 

822977 

456 

177023 

22) 

<  39 

743602 

316 

920352 

140 

823250 

456 

176750 

21  ) 

(  40 

743792 

316 

920268 

140 

823524 

456 

176476 

20  S 

\  41 

9-743982 

316 

9-920184 

140 

9-823798 

456 

10176202 

19) 

)  42 

744171 

316 

920099 

140 

824072 

456 

175928 

18) 

)  43 

744361 

315 

92C015 

140 

824345 

456 

175655 

17) 

)  44 

744550 

315 

919931 

141 

824619 

456 

175381 

16) 

?  45 

744739 

315 

919846 

141 

824893 

456 

175107 

15) 

)  46 

744928 

315 

919762 

141 

825166 

456 

174834 

14  ) 

)  47 

745117 

315 

919677 

141 

825439 

455 

174561 

13) 

7  48 

745306 

314 

919593 

141 

825713 

455 

174287 

12) 

)  49 
J  50 

745494 

314 

919508 

141 

825986 

455 

174014 

11  ) 

745683 

314 

919424 

141 

826259 

455 

173741 

10  \ 

/51 

>  52 

9745871 

314 

9-919339 

141 

9-826532 

455 

10-173468 

9( 

746059 

314 

919254 

141 

826805 

455 

173195 

8 

)   53 

746248 

313 

919109 

141 

827078 

455 

172922 

7  / 

<  S4 

746436 

313 

919085 

141 

827351 

455 

172649 

6) 

/  55 

746624 

313 

919000 

141 

827624 

455 

172376 

5/ 

)  56 

746812 

313 

918915 

142 

827897 

454 

172103 

*i 

)  57 

746999 

313 

918830 

142 

828170 

454 

171830 

3) 

)  53 

747187 

312 

918745 

142 

828442 

454 

171558 

2) 

)  59 

747374 

312 

918659 

142 

828715 

454 

171285 

1 

)  60 

747562 

312 

918574 

142 

828987 

454 

J 71013 

0( 

Cosine 

1 Sine 

1 ^ 

Cotang. 

1 

i~Jfe^ 

ISiJ 

66  Degrees. 


170 


(34  Degrees.)     a    table    OF    LOGARITHMIC 


M. 

|    Sine 

1   D. 

|   Cosine 

1   D. 

Tansr. 

1   D. 

I   Cotang1. 

/ 

0 

9-747562 

312 

9-918574 

142 

9-828987 

454 

10171013 

60  ( 
59  C 

58 

1 

747749 

312 

918489 

142 

829260 

454 

170740 

2 

747036 

312 

918404 

142 

829532 

454 

170468 

i 

748123 

311 

918318 

142 

829805 

454 

170195 

57  ( 

4 

748310 

311 

918233 

142 

830077 

454 

169923 

56  ( 

5 

748497 

311 

918147 

142 

830349 

453 

169651 

55  C 

(3 

748683 

311 

918062 

142 

830621 

453 

169379 

54  I 

7 

748870 

311 

917976 

143 

830893 

453 

169107 

53  ( 

8 

749056 

310 

917891 

143 

831165 

453 

168835 

52  ( 

0 

749243 

310 

917805 

143 

831437 

453 

168563 

51  ( 

10 

749429 

310 

917719 

143 

831709 

453 

168291 

50  ' 

11 

9749615 

310 

9-917634 

143 

9-831981 

453 

10-168019 

49  J 

M 

749801 

310 

917548 

143 

832253 

453 

167747 

48  S 

13 

749987 

309 

917462 

143 

832525 

453 

167475 

47  ) 

14 

750172 

309 

917376 

143 

832796 

453 

167204 

46) 

15 

750358 

309 

917290 

143 

833068 

452 

166932 

45  ) 

1(3 

750543 

309 

917204 

143 

833339 

452 

166661 

44  S 

17 

750729 

309 

917118 

144 

833611 

452 

166389 

43  S 

18 

750914 

308 

917032 

144 

833882 

452 

166118 

42  S 

19 

751099 

308 

916946 

144 

834154 

452 

165846 

41  S 

90 

751284 

308 

916859 

144 

834425 

452 

165575 

40  S 

21 

9751469 

308 

9-916773 

144 

9-834696 

452 

10165304 

39  ; 

22 

751654 

308 

916687 

144 

834967 

452 

165033 

38  > 

33 

751839 

308 

916600 

144 

835238 

452 

164762 

37  ) 

•24 

752023 

307 

916514 

144 

835509 

452 

164491 

36  > 

25 

752208 

307 

916427 

144 

835780 

451 

164220 

35  ) 

2(3 

752392 

307 

916341 

144 

836051 

451 

163949 

34  > 

27 

752576 

307 

910254 

144 

836322 

451 

163678 

33  ) 

28 

752760 

307 

916167 

145 

836593 

451 

163407 

32  ) 

99 

752944 

306 

916081 

145 

836864 

451 

163136 

3i ; 

30 

753128 

306 

915994 

145 

837134 

451 

162866 

30  > 

31 

9753312 

306 

9-915907 

145 

9-837405 

451 

10162595 

29  \ 

32 

753495 

306 

915820 

145 

837675 

451 

162325 

28  ( 

33 

753679 

306 

915733 

145 

837946 

451 

162054 

27  < 

34 

753862 

305 

915646 

145 

838216 

451 

161784 

26  ? 

35 

754046 

305 

915559 

145 

838487 

450 

161513 

25? 

36 

754229 

305 

915472 

145 

838757 

450 

161243 

24  ? 

37 

754412 

305 

915385 

145 

839027 

450 

160973 

23? 

38 

754595 

305 

915297 

145 

839297 

450 

160703 

22? 

3!) 

754778 

304 

915210 

145 

839568 

450 

160432 

21  ? 

40 

754960 

304 

915123 

146 

839838 

450 

160162 

20? 

41 

9755143 

304 

9-915035 

146 

9-840108 

450 

10159892 

19  ( 

42 

755326 

304 

914948 

146 

840378 

450 

159622 

18  ( 

43 

755508 

304 

914860 

146 

840647 

450 

159353 

17  ( 

44 

755690 

304 

914773 

146 

840917 

449 

159083 

16  ( 

45 

755872 

303 

914685 

146 

841187 

449 

158813 

15  < 

46 

756054 

303 

914598 

146 

841457 

449 

158543 

14  ( 

47 

756236 

303 

914510 

146 

841726 

449 

158274 

13  ( 

48 

756418 

303 

914422 

146 

841996 

449 

158004 

12  ( 

49 

756600 

303 

914334 

146 

842266 

449 

157734 

u  S 

50 

756782 

302 

914246 

147 

842535 

449 

157465 

10  ( 

51 

9-756963 

302 

9-914158 

147 

9-842805 

449 

10157195 

9) 

52 

757144 

302 

914070 

147 

843074 

449 

156926 

8) 

53 

757326 

302 

913982 

147 

843343 

449 

156657 

7) 

54 

757507 

302 

913894 

147 

843612 

449 

156388 

c> 

55 

757688 

301 

913806 

147 

843882 

448 

156118 

5S 

96 

757869 

301 

913718 

147 

844151 

448 

155849 

4) 

57 

758050 

301 

913630 

147 

844420 

448 

155580 

3) 

58 

758230 

301 

913541 

147 

844689 

448 

155311 

i 

sy 

758411 

301 

913453 

147 

844958 

448 

155042 

60 

758591 

301 

913365 

147 

845227 

448 

154773 

!   Cosine 

1   Sine 

l^~> 

Cotaiig. 

1    Tang. 

^2 

5 

j  Degree 

S. 

SINKS 

AND  TANGENTS.  (35  Degrees.) 

171 

M. 

S^ 

D. 

Cosine 

D. 

Tariff. 

D.   | 

Cotanff. 

0 

9-758591 

301 

9-913365 

147 

9.845227 

448 

10-154773 

60  j 

1 

758772 

300 

913276 

147 

845496 

448 

154504 

59) 

o 

758952 

300 

913187 

148 

845764 

448 

154236 

58) 

3 

759132 

300 

913099 

148 

84(5033 

448 

153967 

57) 

4 

759312 

300 

913010 

148 

846302 

448 

153698 

56) 

5 

759492 

300 

912922 

148 

846570 

447 

153430 

55) 

6 

759672 

299 

912833 

148 

846839 

447 

153161 

54) 

7 

759852 

299 

912744 

148 

847107 

447 

152893 

53) 
52> 

e 

760031 

299 

912655 

148 

847376 

447 

152624 

9 

760211 

299 

912566 

148 

847644 

447 

152356 

51) 

10 

760390 

299 

912477 

148 

847913 

447 

152087 

50) 

n 

9-760569 

298 

9-912388 

148 

9.848181 

447 

10151819 

49? 

12 

760748 

298 

912299 

149 

848449 

447 

151551 

48? 

13 

760927 

298 

912210 

149 

848717 

447 

151283 

47? 

14 

761106 

298 

912121 

149 

848986 

447 

151014 

46? 

15 

761285 

298 

912031 

149 

849254 

447 

150746 

45  ? 

16 

761464 

298 

911942 

149 

849522 

447 

150478 

44? 

17 

761642 

297 

911853 

149 

849790 

446 

150210 

43? 

18 

761821 

297 

911763 

149 

850058 

446 

149942 

42? 

19 

761999 

297 

911674 

149 

850325 

446 

149675 

41? 

20 

762177 

297 

911584 

149 

850593 

446 

149407 

40? 

21 

9762356 

297 

9-911495 

149 

9850861 

446 

10-149139 

39  ( 

22 

762534 

296 

911405 

149 

851129 

446 

148871 

38  < 

23 

762712 

296 

911315 

150 

851396 

446 

148604 

37  ( 

24 

762889 

296 

911226 

150 

851664 

446 

148336 

36  ( 

25 

763067 

296 

911136 

150 

851931 

446 

148069 

35( 

26 

763245 

296 

911046 

150 

852199 

446 

147801 

34  ( 

27 

763422 

296 

910956 

150 

852466 

446 

147534 

33  ( 

28 

763600 

295 

910866 

150 

852733 

445 

147267 

32  ( 

29 

763777 

295 

910776 

150 

853001 

445 

146999 

31  ( 

30 

763954 

295 

910686 

150 

853268 

445 

146732 

30  ( 

31 

9-7641 31 

295 

9-910596 

150 

9-853535 

445 

10-146465 

29  J 

32 

764308 

295 

910506 

150 

853802 

445 

146198 

28  S 

33 

764485 

294 

910415 

150 

854069 

445 

145931 

27  ) 

34 

764662 

294 

910325 

151 

854336 

445 

145664 

26  S 

35 

764838 

294 

910235 

151 

854603 

445 

145397 

25  S 

36 

765015 

294 

910144 

151 

854870 

445 

145130 

24  \ 

37 

765191 

294 

910054 

151 

855137 

445 

144863 

23$ 

38 

765367 

294 

909963 

151 

855404 

445 

144596 

22  \ 
21  ' 

39 

765544 

293 

909873 

151 

855671 

444 

144329 

40 

765720 

293 

909782 

151 

855938 

444 

144062 

20  <, 

41 

9-765896 

293 

9-909691 

151 

9-856204 

444 

10143796 

29  S 

42 

766072 

2<>3 

909601 

151 

856471 

444 

143529 

18) 

43 

766247 

293 

909510 

151 

856737 

444 

143263 

17) 

44 

766423 

293 

909419 

151 

857004 

444 

142996 

16) 

45 

766598 

292 

909328 

152 

857270 

444 

142730 

15) 

46 

766774 

292 

909237 

152 

857537 

444 

142463 

14) 

47 

766949 

292 

909146 

152 

857803 

444 

142197 

13) 

48 

767124 

292 

909055 

152 

858069 

444 

141931 

12) 

49 

767300 

292 

908964 

152 

858336 

444 

141664 

11  ) 

50 

767475 

291 

908873 

152 

858602 

443 

141398 

10  j 

51 

9-767649 

291 

9-908781 

152 

9-858863 

443 

10-1411r2 

9? 

52 

767824 

291 

908690 

152 

859134 

443 

140866 

8? 

53 

767999 

291 

908599 

152 

859400 

443 

140600 

7) 

54 

768173 

291 

908507 

152 

859666 

443 

140334 

6) 

55 

768348 

290 

908416 

153 

859932 

443 

140068 

5? 

56 

768522 

290 

908324 

153 

860198 

443 

139802 

4 

57 

768697 

290 

908233 

153 

860464 

443 

139536 

3) 

58 

768871 

290 

908141 

153 

860730 

443 

139270 

S( 

59 

769045 

290 

908049 

153 

860995 

443 

139005 

)  ^ 

769219 

290 

907958 

153 

861261 

443 

138739 

0) 

i 

1   Cosine 

1 

|    Sine 

1 

|   Cotang. 

1 

I   Tan?. 

M.? 

54  Degrees. 


172 


(36  Degrees.)     A    TABLE    OP    LOGARITHMIC 


[m/ 

Sine 

1   D- 

Cosine 

D. 

Tang. 

D. 

Cotang. 

(  ° 

9-769219 

290 

9-907958 

153 

9-861261 

443 

10-138739 

60  \ 

?   1 

769393 

289 

907866 

153 

861527 

443 

138473 

59  ( 

(  2 

769566 

289 

907774 

153 

861792 

442 

138208 

58  ( 

(  3 

769740 

289 

907682 

353 

862058 

442 

137942 

57  < 

(  4 

'09913 

289 

907590 

153 

862323 

442 

137677 

56  ( 

5 

770087 

289 

907498 

153 

862589 

442 

137411 

55  ( 

(  6 

770260 

288 

907406 

153 

862854 

442 

137146 

54  ( 

7 

770433 

288 

907314 

154 

863119 

442 

136881 

53  ( 

8 

770606 

288 

907222 

154 

863385 

442 

136615 

52  ( 

C  9 

770779 

288 

907129 

154 

863650 

442 

136350 

51  < 

! 10 

770952 

288 

907037 

154 

863915 

442 

136085 

50  j 

\ n 

9-771125 

288 

9-906945 

154 

9-864180 

442 

10135820 

49  I 

12 

771298 

287 

906852 

154 

864445 

442 

135555 

48  ( 

13 

771470 

287 

906760 

154 

864710 

442 

135290 

47  < 

14 

771643 

287 

906667 

154 

864975 

441 

135025 

46  ( 

15 

771815 

287 

906575 

154 

865240 

441 

134760 

45  ( 

(  10 

771987 

287 

906482 

154 

865505 

441 

134495 

44  ( 

(  17 

772159 

287 

906389 

155 

865770 

441 

134230 

43  ( 

)  18 

772331 

286 

906296 

155 

866035 

441 

133965 

42  ( 

)  19 

772503 

286 

90G204 

155 

866300 

441 

133700 

41  ( 

J  20 

772675 

286 

906111 

155 

866564 

441 

133436 

40 

}  21 

9-772847 

286 

9-906018 

155 

9-866829 

441 

10133171 

39  { 

)  22 

773018 

286 

905925 

155 

867094 

441 

132906 

38  S 

)  23 

773190 

286 

905832 

155 

867358 

441 

232642 

37  > 

)  24 

773361 

285 

905739 

155 

867623 

441 

132377 

36  } 
35  < 

s 

32  ( 

)  25 

773533 

285 

905645 

155 

867887 

441 

132113 

}  28 

773704 

285 

905552 

155 

868152 

440 

131848 

;  27 

773875 

285 

905459 

155 

868416 

440 

131584 

)  28 

774046 

285 

905366 

156 

868680 

440 

131320 

)  29 

774217 

285 

905272 

156 

868945 

440 

131055 

31  ( 

)  30 

774388 

284 

905179 

156 

869209 

440 

130791 

30  < 

(  31 

9-774558 

284 

9-9050135 

156 

9-869473 

440 

10130527 

29  ) 

(  32 

774729 

284 

904992 

156 

869737 

440 

130263 

28  S 

(  33 

774899 

284 

904898 

156 

870001 

440 

129999 

27  ) 

(  34 

775070 

284 

904804 

156 

870265 

440 

129735 

26  S 

(  35 

775240 

284 

904711 

156 

870529 

440 

129471 

25  > 

(  36 

775410 

283 

904617 

156 

870793 

440 

129207 

24  ) 

(  37 

775580 

283 

904523 

156 

871057 

440 

128943 

23  S 

(  38 

775750 

283 

904429 

157 

871321 

440 

128679 

22  S 

(  39 

775920 

283 

904335 

157 

871585 

440 

128415 

21  ( 

j  40 

776090 

283 

904241 

157 

871849 

439 

128151 

20  j 

J  41 

9-776259 

283 

9-904147 

157 

9-872112 

439 

10-127888 

19  ) 

)  42 

776429 

282 

904053 

157 

872376 

439 

127624 

18  ) 

)  43 

776598 

282 

903959 

157 

872640 

439 

127360 

17  ) 

)   44 

776768 

282 

903864 

157 

872903 

439 

127097 

16  ) 

\  45 

776937 

282 

903770 

157 

873167 

439 

126833 

15  ) 

\  46 

777106 

282 

903676 

157 

873430 

439 

126570 

14  > 

)  47 

777275 

281 

903581 

157 

873694 

439 

126306 

13  ) 

S  48 

777444 

281 

903487 

157 

873957 

439 

126043 

12) 

S  49 

777613 

281 

903392 

158 

874220 

439 

125780 

11 
10  < 

<  50 

777781 

281 

903298 

158 

874484 

439 

125516 

}  51 

9777950 

281 

9-903203 

158 

9-874747 

439 

10-125253 

9) 

)  52 

778119 

281 

903108 

158 

875010 

439 

124990 

8  ( 

)  53 

778287 

280 

903014 

158 

875273 

438 

124727 

7  I 

)  54 

778455 

280 

902919 

158 

875536 

438 

124464 

6  ( 

)  55 

778624 

280 

902824 

158 

875800 

438 

124200 

5 

)  56 

778792 

280 

902729 

158 

870063 

438 

123937 

4' 

)  57 

778960 

280 

902634 

158 

876326 

438 

123674 

3< 

)  58 

779128 

280 

902539 

159 

876589 

438 

123411 

2) 

)  59 

779295 

279 

902444 

159 

876851 

438 

123149 

1  ) 

j  60 

779463 

279 

902349 

159 

877114 

438 

122886 

0 

j 

Cosine 

Smj 

Cotang. 

Tang. 

M.  j 

53  Degrees. 


INES    AND    tangents.     (37  Degrees.) 


173 


p5^ 

1  o 

sine 

D. 

Cosine 

D. 

Tan* 

D. 

Cotang.   |     ) 

9-779463 

279 

9902349 

159 

9-877114 

438 

10-122886 

60  ( 

)   i 

779631 

279 

902253 

159 

877377 

438 

122623 

59  ( 

l    2 

779798 

279 

902158 

159 

877640 

438 

122360 

58  ) 

f     3 

779966 

279 

902063 

159 

877903 

438 

122097 

57  / 

)  4 

780133 

279 

901967 

159 

878105 

438 

121835 

56 ; 

55  > 

/  5 

780300 

278 

901872 

159 

878428 

438 

121572 

>  G 

780467 

278 

90J"76 

159 

878691 

438 

121309 

54  > 

/  7 

780634 

278 

901681 

159 

878953 

437 

121047 

53  ) 

)  8 

780801 

278 

901585 

159 

879216 

437 

120784 

52  ) 

)  9 

780968 

278 

901490 

159 

879478 

437 

120522 

51  ) 

1  10 

781134 

278 

901394 

160 

879741 

437 

120259 

50  J 

c  u 

9-781301 

277 

9-901298 

160 

9-880003 

437 

10119997 

49  ) 

(  12 

781468 

277 

901202 

160 

880265 

437 

119735 

48  } 

I   13 

781634 

277 

901106 

160 

880528 

437 

119472 

47  > 

(  I4 

781800 

277 

901010 

160 

880790 

437 

119210 

46  ) 

(  I5 

781966 

277 

900914 

160 

881052 

437 

118948 

45  ) 

(  16 

^32132 

277 

900818 

160 

881314 

437 

118686 

44 

c  n 

782298 

276 

900722 

160 

881576 

437 

118424 

43 

)  18 

782464 

276 

900626 

160 

881839 

437 

118161 

42  ) 

>  19 

782630 

276 

900529 

160 

882101 

437 

117899 

41  ) 

>  2a 

782796 

276 

900433 

161 

882363 

436 

117637 

40  ) 

(  21 

9-782961 

276 

9-900337 

161 

9882625 

436 

10117375 

39  j 

'  22 

783127 

276 

900240 

161 

882887 

436 

117113 

38  ) 

<  23 

783292 

275 

900144 

161 

883148 

436 

116852 

37  ) 

<  24 

783458 

275 

900047 

161 

883410 

436 

116590 

36  ) 

<  25 

783623 

275 

899951 

161 

883672 

436 

116328 

35  ) 

I   26 

783788 

275 

899854 

161 

883934 

436 

116066 

34  ) 

<  27 

783953 

275 

899757 

161 

884196 

•436 

115804 

33  ) 

?  28 

784118 

275 

899660 

161 

884457 

436 

115543 

32  > 

<  29 

784282 

274 

899564 

161 

884719 

436 

115281 

31  S 

?  30 

784447 

274 

899467 

162 

884980 

436 

115020 

30  S 

S  31 

9-784612 

274 

9-899370 

162 

9-885242 

436 

10114758 

29  ) 

C  32 

784776 

274 

899273 

162 

885503 

436 

114497 

28  ) 

<  33 

784941 

274 

899176 

162 

885765 

436 

114235 

27  ) 

S  34 

785105 

274 

899078 

162 

886026 

436 

113974 

26  ) 

(  35 

785269 

273 

898981 

162 

886288 

436 

113712 

25  ) 

<  36 

785433 

273 

898884 

162 

886549 

435 

113451 

24  > 

(  37 

785597 

273 

898787 

162 

886810 

435 

113190 

23  ) 

<  38 

785761 

273 

898689 

162 

887072 

435 

112928 

22  ) 

<  39 

785925 

273 

898592 

162 

887333 

435 

1 J 2667 

21  ) 

j  40 

786089 

273 

898494 

163 

887594 

435 

112406 

20  j 

)  41 

9-786252 

272 

9-898397 

163 

9-887855 

435 

10112145 

19  ) 

(  42 

786416 

272 

898299 

163 

888116 

435 

111884 

18  / 

<  43 

786579 

272 

898202 

163 

888377 

435 

111623 

17  / 

<  44 

786742 

272 

898104 

163 

888639 

435 

111361 

16  ) 

(  45 

786906 

272 

898006 

163 

888900 

435 

111100 

15  ) 

<  46 

787069 

272 

897908 

163 

889160 

435 

110840 

14  ) 

(  47 

787232 

271 

897810 

163 

889421 

435 

110579 

13  > 

<  48 

787395 

271 

897712 

163 

889682 

435 

110318 

12  ) 

<  49 

787557 

271 

897614 

163 

889943 

435 

110057 

11  ? 

(  50 

787720 

271 

897516 

163 

890204 

434 

119796 

10  ) 

>  51 

9-787883 

271 

9-897418 

164 

9-890465 

434 

10109535 

9  ( 

S  52 

788045 

271 

897320 

164 

890725 

434 

109275 

8  ( 

J  53 

788208 

271 

897222 

164 

890986 

434 

109014 

7  I 

;  54 

788370 

270 

897123 

164 

891247 

434 

108753 

6  ? 

)  55 

788532 

270 

897025 

164 

891507 

434 

108493 

5  ( 

)   56 

788694 

270 

896926 

164 

891768 

434 

108232 

4  ( 

)  57 

788856 

270 

896828 

164 

892028 

434 

107972 

3{ 
2  / 

)  58 

789018 

270 

896729 

164 

892289 

434 

107711 

)  59 

789180 

270 

896631 

164 

892549 

434 

107451 

1  ) 

J  60 

789342 

269 

896532 

164 

892810 

434 

107190 

0  ) 

Ly- 

Cosine 

Sine 

-^~^~ 

Cotang. 

•N.O^-  -^>_ 

Tang. 

^mJ 

52  Degrees. 


174 


(38  Degrees.)    a  table   of   logarithmic 


CiC 

Sine 

D.   | 

Cosine 

D. 

Tang.   | 

^dT^ 

Cotp.ng 

( 

(  o 

9789342 

269 

9896532 

164 

9  892810 

434 

10107190 

60  ( 

)  i 

789504 

269 

896433 

165 

893070 

434 

106930 

59  S 

;  2 

789665 

269 

896335 

165 

893331 

434 

106669 

58  S 

;  3 

789827 

269 

896236 

165 

893591 

434 

106409 

57  S 

)  4 

789988 

269 

896137 

165 

893851 

434 

106149 

56  ( 

)   5 

790149 

269 

896038 

165 

894111 

434 

105889 

55  , 

\   6 

790310 

268 

895939 

165 

894371 

434 

105629 

54 
53  \ 

J  7 

790471 

268 

895840 

165 

894632 

433 

105368 

/  8 
\    9 

790632 

268 

895741 

165 

894892 

433 

105108 

52  \ 

790793 

268 

895641 

165 

895152 

433 

104848 

51  < 

,10 

790954 

268 

895542 

165 

895412 

433 

104588 

50  j 

)  11 

9  791115 

268 

9-895443 

166 

9-895672 

433 

10104328 

49  J 

)  12 

791275 

267 

895343 

166 

895932 

433 

104068 

48  ) 

)  13 

791436 

267 

895244 

166 

896192 

433 

103808 

47  ) 

)  14 

791596 

267 

895145 

1(56 

896452 

433 

103548 

46  ) 

)   J5 

791757 

267 

895045 

166 

896712 

433 

103288 

45  ) 

i   16 

791917 

267 

894945 

166 

896971 

433 

103029 

44  > 

)  17 

792077 

267 

894846 

166 

897231 

433 

102769 

43) 

I   18 

792237 

266 

894746 

166 

897491 

433 

102509 

42) 

)   19 

792397 

266 

894646 

166 

897751 

433 

102249 

41  ) 

)  20 

792557 

266 

894546 

166 

898010 

433 

101990 

40  j 
39/ 

(  21 

9792716 

266 

9-894446 

167 

9-898270 

433 

10101730 

(  22 

792876 

2(56 

894346 

167 

898530 

433 

101470 

38/ 

(  23 

793035 

266 

894246 

167 

898789 

433 

101211 

37) 

(  24 

793195 

265 

894146 

167 

899049 

432 

100951 

36  > 

(  25 

793354 

265 

894046 

167 

899308 

432 

100692 

35) 

(  26 

793514 

265 

893946 

167 

899568 

432 

100432 

34) 

(  27 

793673 

265  ' 

893846 

167 

899827 

432 

J00173 

33) 

(  28 

793832 

265 

893745 

167 

900086 

432 

099914 

32) 

f  29 

793991 

265 

893645 

167 

900346 

432 

099654 

31  ) 

(  30 

794150 

264 

893544 

167 

900605 

432 

099395 

30 ) 

I  31 

9794308 

264 

9-893444 

168 

9-900864 

432 

10099136 

29( 

33 

794467 

264 

893343 

168 

901124 

432 

098876 

28( 

33 

794626 

264 

893243 

168 

901383 

432 

098617 

27  C 

)  34 

794784 

264 

893142 

168 

901642 

432 

098358 

26  I 

)   35 

794942 

264 

893041 

168 

901901 

432 

098099 

25  < 

S  36 

795101 

264 

892940 

168 

902160 

432 

097840 

24  ( 

(  37 

795259 

264 

892839 

168 

902419 

432 

097581 

23  I 

S   38 

795417 

263 

892739 

168 

902679 

432 

097321 

22  < 

(  39 

795575 

263 

892638 

168 

902938 

432 

097062 

21  ( 

>  40 

795733 

263 

892536 

168 

903197 

431 

096803 

20  j 

1  41 

9  795891 

263 

9-892435 

169 

9-903455 

431 

10  096545 

19 

)  42 

796049 

263 

892334 

169 

903714 

431 

096286 

18  S 

)  43 

796206 

263 

892233 

169 

903973 

431 

096027 

17  < 

)  44 

796364 

262 

892132 

169 

904232 

431 

095768 

16  ( 

)  45 

796521 

262 

892030 

169 

904491 

431 

095509 

15  < 

)  46 

796679 

262 

891929 

169 

904750 

431 

095250 

14  ( 

)  47 

796836 

262 

891827 

169 

905008 

431 

094992 

13  < 

)  48 

796993 

262 

891726 

169 

905267 

431 

094733 

12 

)  49 

797150 

261 

891624 

169 

905526 

431 

094474 

11  ! 

J  50 

797307 

261 

891523 

170 

905784 

431 

094216 

10  ( 

(  51 

9797464 

261 

9-891421 

170 

9-906043 

431 

10093957 

9! 

(  52 

797621 

261 

891319 

170 

906302 

431 

093698 

8) 

(  53 

797777 

261 

891217 

170 

906560 

431 

093440 

7) 

)  54 

797934 

261 

891115 

170 

906819 

431 

093181 

6  ) 

/  55 

798091 

261 

891013 

170 

907077 

431 

092923 

5; 

(   56 

798247 

261 

890911 

170 

907336 

431 

092664 

4 ; 

<  57 

798403 

260 

890809 

170 

907594 

431 

092406 

3) 

/  58 

798360 

260 

890707 

170 

907852 

431 

092148 

2) 

/  59 

798716 

260 

890605 

170 

908111 

430 

091889 

15 

?  60 

798872 

260 

890503 

170 

908369 

430 

091631 

I  0 

I       Cosine 


51  Degrees. 


bines   AND   tang K NTS,    (30  Degrees.) 


175 


>  o 

9-798872 

260 

9-890503 

170 

9-908309 

430 

10-091631 

60? 

)   1 

799028 

260 

890400 

171 

908628 

430 

091372 

59? 

2 

799184 

260 

890298 

171 

908886 

430 

091114 

58/ 

3 

799339 

259 

890195 

171 

909144 

430 

090856 

57  ) 

4 

799495 

259 

890093 

171 

909402 

430 

090598 

56  ) 

)  5 

799051 

259 

889990 

171 

909600 

430 

090340 

55  > 

)  <> 

799806 

259 

889888 

171 

909918 

430 

090082 

51? 

7 

799962 

259 

889785 

171 

910177 

430 

089823 

53  ) 

>  8 

800117 

259 

889682 

171 

910435 

430 

089565 

52? 

>  9 

800272 

258 

889579 

171 

910693 

430 

089307 

51 1 

10 

800427 

258 

889477 

171 

910951 

430 

089049 

50) 
49  I 

>  11 

9800582 

258 

9-889374 

172 

9911209 

430 

10-088791 

12 

800737 

258 

889271 

172 

9114G7 

430 

088533 

48? 

13 

800892 

258 

889168 

172 

911724 

430 

088276 

47  ( 

14 

801047 

258 

889064 

172 

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430 

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46  < 
45  I 

15 

801201 

258 

888961 

172 

912240 

430 

087760 

>   10 

801356 

257 

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172 

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430 

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44  ( 

>  17 

801511 

257 

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172 

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430 

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43  < 

i  18 

801665 

257 

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172 

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429 

C8C986 

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i  19 

801819 

257 

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172 

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429 

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20 

801973 

257 

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173 

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429 

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257 

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256 

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429 

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1  23 

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256 

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173 

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256 

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429 

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25 

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256 

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173 

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429 

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35$ 

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256 

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173 

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429 

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34  S 

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256 

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173 

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28 

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256 

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29 

803357 

255 

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173 

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429 

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30 

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9-803664 

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429 

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174 

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174 

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428 

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22  ) 

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254 

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174 

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428 

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805039 

254 

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175 

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428 

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J  41 

9-805191 

254 

9-886257 

175 

9-918934 

428 

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19? 

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253 

886152 

175 

919191 

428 

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18? 

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805495 

253 

886047 

175 

919448 

428 

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17  ( 

(  44 

805647 

253 

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175 

919705 

428 

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16) 

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253 

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175 

919962 

428 

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15) 

(  46 

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253 

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175 

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428 

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14  » 

(  47 

806103 

253 

885627 

175 

920476 

428 

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13) 

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(  49 

806254 

253 

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175 

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428 

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12) 

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252 

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175 

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428 

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806557 

252 

885311 

176 

921247 

428 

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10  ) 

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252 

9-885205 

176 

9921503 

428 

10*078497 

of 

)  52 

806860 

252 

885100 

176 

921760 

428 

078240 

8S 

)   53 

807011 

252 

884994 

176 

922017 

428 

077983 

7S 

54 

807163 

252 

884889 

176 

922274 

428 

077726 

6S 

55 

S  58 
<57 

807314 

252 

884783 

176 

922530 

428 

077470 

5! 

807465 

251 

884677 

176 

922787 

428 

077213 

4 

807615 

251 

884572 

176 

923044 

428 

076956 

3 

)  58 

807766 

251 

884466 

176 

923300 

428 

076700 

2S 

)  59 

807917 

251 

884360 

176 

923557 

427 

076443 

l\ 

}60 

808067 

251 

884254 

177 

923813 

427 

076187 

°i> 

I 

;   Coeine 

J^ Sine 

I 

|   Cotang\ 

l^~^ 

1~-32T£~- 

ikj 

V-^. 

1 

0  Decree 

S. 

^-^~/ 

176 


(40 Degrees.)     a    table    op    logarithmic 


<  M.    |       Sine 


I      D. 


Cosine       J 


I       Tang. | 


I Cotanjf.      | 


0 

9-808067 

251 

9-884254 

177 

9-923813 

427 

10-076187 

60  / 

1 

808218 

251 

884148 

177 

924070 

427 

075930 

59  I 

)  2 

808368 

251 

884042 

177 

924327 

427 

075673 

58  / 

)  3 

808519 

250 

883936 

177 

924583 

427 

075417 

57  ) 

)  4 

808669 

250 

883829 

177 

924840 

427 

075160 

56  I 

)    5 

808819 

250 

883723 

177 

925096 

427 

074904 

55  / 

)    6 

808969 

250 

883617 

177 

925352 

427 

074648 

54  ; 

>  7 

809119 

250 

883510 

177 

925609 

427 

074391 

53  / 

)    8 

809269 

250 

883404 

177 

925865 

427 

074135 

52  / 

)  9 

809419 

249 

883297 

178 

926122 

427 

073878 

51  / 

S  10 

809569 

249 

883191 

178 

926378 

427 

073622 

50  / 

>  11 

9809718 

249 

9-883084 

178 

9-926634 

427 

10073366 

49  ( 

)  J2 

809868 

249 

882977 

178 

926890 

427 

073110 

48  <, 

)  13 

810017 

249 

882871 

178 

927147 

427 

072853 

47  ( 

)  14 

810167 

249 

882764 

178 

927403 

427 

072597 

46  < 

>  15 

810316 

248 

882657 

178 

927659 

427 

072341 

45  < 

I   16 

810465 

248 

882550 

178 

927915 

427 

072085 

44  < 

I   17 

810614 

248 

882443 

178 

928171 

427 

071829 

43  ( 

)  18 

810763 

248 

882336 

179 

928427 

928683 

427 

071573 

42  ( 

)  19 

810912 

248 

882229 

179 

427 

071317 

41  ( 

?  20 

811061 

248 

882121 

179 

928940 

427 

071060 

40  ( 

(  21 

9-811210 

248 

9-882014 

179 

9-929196 

427 

10070804 

39  ) 

(  22 

811358 

247 

881907 

179 

929452 

427 

070548 

38  ) 

(  23 

811507 

247 

881799 

179 

929708 

427 

070292 

37  ) 

(  24 

811655 

247 

881692 

179 

929964 

426 

070036 

36  ) 

(  25 

811804 

247 

881584 

179 

930220 

426 

069780 

35  ) 

(  26 

811952 

247 

881477 

179 

930475 

426 

069525 

34  S 

(27 

812100 

247 

881369 

179 

930731 

426 

069269 

33  S 

(  28 

8i2248 

247 

881261 

180 

930987 

426 

069013 

32  S 

<  29 

812396 

246 

881153 

180 

931243 

426 

068757 

31  S 

(  30 

812544 

246 

881046 

180 

931499 

426 

068501 

30  j 

(  31 

9812692 

246 

9-880938 

180 

9  931755 

426 

10068245 

29  J 

)  32 

812840 

246 

880830 

180 

932010 

426 

067990 

28  ) 

)  33 

812988 

246 

880722 

180 

932266 

426 

067734 

27  ) 

)34 

813135 

246 

880613 

180 

932522 

426 

067478 

26  > 

>  35 

813283 

246 

880505 

180 

932778 

426 

067222 

25  ) 

)  36 

813430 

245 

880397 

180 

933033 

426 

066907 

24) 

)  37 

813578 

245 

880289 

181 

933289 

426 

066711 

23  ) 

>  38 

813725 

245 

880180 

181 

933545 

426 

066455 

22  ) 

)  39 

813872 

245 

880072 

181 

933800 

426 

066200 

21  j 

)  40 

814019 

245 

879963 

181 

934056 

426 

065944 

20  S 

J  41 

9-814166 

245 

9-879855 

181 

9934311 

426 

10065689 

19  ) 

(42 

814313 

245 

879746 

181 

934567 

426 

065433 

18  ) 

43 

814460 

244 

879637 

181 

934823 

426 

065177 

17  ) 

44 

814607 

244 

879529 

181 

935078 

426 

064922 

16  ) 

(  45 

814753 

244 

879420 

181 

935333 

426 

064667 

15  ) 

(46 

814900 

244 

879311 

181 

935589 

426 

064411 

14  ) 

<  47 

815046 

244 

879202 

182 

935844 

426 

064156 

13  ) 

(48 

815193 

244 

879093 

182 

936100 

426 

063900 

12  ) 

(49 

815339 

244 

878984 

182 

936355 

426 

063645 

11  / 

(50 

815485 

243 

878875 

182 

936610 

426 

063390 

10  J 

)51 

9-815631 

243 

9-878766 

182 

9-936866 

425 

10-063134 

9  ( 

)  52 

815778 

243 

878656 

182 

937121 

425 

062879 

8  ( 

)  53 

815924 

243 

878547 

182 

937376 

425 

062624 

7  ( 

)  54 

816069 

243 

878438 

182 

937632 

425 

062368 

6  / 

S  55 

816215 

243 

878328 

182 

937887 

425 

062113 

5  ) 

)  56 

816361 

243 

878219 

183 

938142 

425 

061858 

4  ( 

>  57 

816507 

242 

878109 

183 

938398 

425 

061002 

3  ) 

)  58 

816652 

242 

877999 

183 

938653 

425 

061347 

2  ) 

)59 

816798 

242 

877890 

183 

938908 

425 

061092 

1  ) 

S60 

816943 

242 

877780 

183 

939163 

425 

060837 

0  j 

r 

|   Cosine 

I 

|    Sine 

1 

|   Cotan^. 

1 

I    Tan*. 

1  M.  1 

49  Degrees. 


SINES    4ND   TANGENTS.     (41  Degrees.) 


177 


|         D.       |       Cosine 


|       Ttog         |       D.       |      Cotang.        | 


)  0 

9-816943 

242 

9877780 

183 

9939163 

425 

10060837 

60 

(  1 

817088 

242 

877670 

183 

939418 

425 

060582 

59 

)    2 

817233 

242 

877560 

183 

939673 

425 

060327 

58 

)  3 

817379 

242 

877450 

183 

939928 

425 

060072 

57 

',    4 

817524 

241 

877340  - 

183 

940183 

425 

059817 

56 

)    5 

817068 

241 

877230 

184 

940438 

425 

059562 

55 

S    6 

817813 

241 

877120 

184 

940694 

425 

059306 

54 

S    7 

817958 

241 

877010 

184 

940949 

425 

059051 

53 

4  8 

818103 

241 

876899 

184 

941204 

425 

058796 

52 

S  9 

818247 

241 

876789 

184 

941458 

425 

058542 

51 

<  10 

818392 

241 

876678 

184 

941714 

425 

058286 

50 

5  n 

9-818536 

240 

9  876508 

184 

9-941968 

425 

10-058032 

49 

>13 

<  13 

818681 

240 

876457 

184 

942223 

425 

057777 

48 

818825 

240 

876347 

184 

942478 

425 

057522 

47 

\  14 

818969 

240 

876236 

185 

942733 

425 

057267 

46 

SJ5 

819113 

240 

876125 

185 

942988 

425 

057012 

45 

(l6 

819257 

240 

876014 

185 

943243 

425 

056757 

44 

S  17 

819401 

240 

875904 

185 

943498 

425 

056502 

43' 

)  18 

819545 

239 

875793 

185 

943752 

425 

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42 

S j9 

819689 

239 

875682 

185 

944007 

425 

055993 

41 

^20 

819832 

239 

875571 

185 

944262 

425 

055738 

40' 

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9-819976 

239 

9-875459 

185 

9-944517 

425 

10055483 

39 

;22 

820120 

239 

875348 

185 

944771 

424 

055229 

38 

>23 

820263 

239 

875237 

185 

945026 

424 

054974 

37 

)24 

820406 

239 

875126 

186 

945281 

424 

054719 

36 

25 

820550 

238 

875014 

186 

945535 

424 

054465 

35) 

)26 

820693 

238 

874903 

186 

945790 

424 

054210 

34) 

}27 

820836 

238 

874791 

186 

946045 

424 

053955 

33) 

)28 

820979 

238 

874680 

186 

946299 

424 

053701 

32) 

S  29 

821122 

238 

874568 

186 

946554 

424 

053446 

31) 

J  30 

821265 

238 

874456 

186 

946808 

424 

053192 

30  ) 

1  31 

9821407 

238 

9-874344 

186 

9947063 

424 

10052937 

29  J 

>32 

821550 

238 

874232 

187 

947318 

424 

052682 

28  ( 

X33 

821693 

237 

874121 

187 

947572 

424 

052428 

27  ( 

)34 

821835 

237 

874009 

187 

947826 

424 

052174 

26  ( 

)35 

821977 

237 

873896 

187 

948081 

424 

051919 

25  l 

)36 

822120 

237 

873784 

187 

948336 

424 

051664 

24  I 

)37 

822262 

237 

873672 

187 

948590 

424 

051410 

23  ( 

)38 

822404 

237 

873560 

187 

948844 

424 

051156 

22  ( 

)39 

822546 

237 

873448 

187 

949099 

424 

050901 

21  ) 

)40 

822688 

236 

873335 

187 

949353 

424 

050647 

20 

J  41 

9-822830 

236 

9-873223 

187 

9-949607 

424 

10.050393 

19 

(42 

822972 

236 

873110 

188 

949862 

424 

050138 

18  S 

?43 

823114 

236 

872998 

188 

950116 

424 

049884 

17  ( 

/44 

823255 

236 

872885 

188 

950370 

424 

049630 

16  > 

?45 

823397 

236 

872772 

188 

950625 

424 

049375 

15) 

?46 

823539 

236 

872659 

188 

950879 

424 

049121 

14  \ 

)47 

823680 

235 

872547 

188 

951133 

424 

048867 

13  ( 

>48 

823821 

235 

872434 

188 

951388 

424 

048612 

12  > 

■49 

823963 

235 

872321 

188 

951642 

424 

048358 

U) 

>50 

824104 

235 

872208 

188 

951896 

424 

048104 

10  J 

51 

9-824245 

235 

9-872095 

189 

9-952150 

424 

10047850 

9  ; 

52 

824386 

235 

871981 

189 

952405 

424 

047595 

?  ) 

53 

824527 

235 

871868 

189 

952659 

424 

047341 

7  ? 

54 

824668 

234 

871755 

189 

952913 

424 

047087 

6) 

(  55 

824808 

234 

871641 

189 

953167 

423 

046833 

Si 

(50 

824949 

234 

871528 

189 

953421 

423 

046579 

57 

825090 

234 

871414 

189 

953675 

423 

046325 

3) 

58 

825230 

234 

871301 

189 

953929 

423 

046071 

2  ) 

(59 

825371 

234 

871187 

189 

954183 

423 

0458 J 7 

1  ) 

(60 

825511 

234 

871073 

190 

954437 

423 

045563 

w 

I  I       Cotaug. 

48  Degrees. 


i  M.r 


178 


(-12  Degrees  )     a  TABLE   OF   LOGARITHMIC 


smT 

Sine 

1   D. 

|  Cosine 

1   I>- 

1    Tan,?. 

1   D- 

|   Cotsng. 

]~i 

J  0 

9-825511 

234 

9-871073 

|   190 

9-954437 

423 

10-045563 

60  { 

1 

825651 

233 

870960 

(  190 

954691 

423 

045309 

59  { 

58  ( 

)  2 

825791 

233 

870846 

1  190 

954945 

423 

045055 

3 
(    * 

(     5 

825931 

233 

870732 

190 

955200 

423 

044800 

57  ( 

826071 

233 

870618 

190 

955454 

423 

044546 

56  ( 

826211 

233 

870504 

190 

955707 

423 

044293 

55  ( 

6 

826351 

233 

870390 

190 

955961 

423 

044039 

54  I 

7 

826491 

233 

870276 

190 

956215 

423 

043785 

53  I 

8 
9 

826631 

233 

870161 

190 

956469 

423 

043531 

52  ( 

826770 

232 

870047 

191 

956723 

423 

043277 

51  ( 

826910 

232 

869933 

191 

956977 

423 

043023 

50  j 

^ n 

9-827049 

232 

9-869818 

191 

9-957231 

423 

10-042769 

49  j 

12 

827189 

232 

869704 

191 

957485 

423 

042515 

48  \ 

I   13 

827328 

232 

869589 

191 

957739 

423 

042261 

47  S 

'  14 

827467 

232 

869474 

191 

957993 

423 

042007 

46  J 

t  IS 
<  16 

17 

.18 
19 
20 

1  21 

827606 

232 

869360 

191 

958246 

423 

041754 

45  \ 

827745 

232 

869245 

191 

958500 

423 

041500 

44  ) 

827884 

231 

869130 

191 

958754 

423 

041246 

43  S 

828023 

231 

869015 

192 

959008 

423 

040992 

42  \ 

828162 

231 

868900 

192 

959262 

423 

040738 

41  ; 

828301 

231 

868785 

192 

959516 

423 

040484 

40  • 

0828439 

231 

9-868670 

192 

9-959769 

423 

10040231 

39  ) 

'.22 

828578 

231 

868555 

192 

900023 

423 

039977 

38  > 

23 

828716 

231 

868440 

192 

9G0277 

423 

039723 

37  ) 

1  24 

828855 

230 

868324 

192 

960531 

423 

039469 

36  ) 

)  25 

828993 

230 

868209 

192 

960784 

423 

039216 

35  ) 

26 

829131 

230 

868093 

192 

961038 

423 

038962 

34  ) 

',27 

829269 

230 

867978 

193 

961291 

423 

038709 

33  ; 
32  / 

'  28 

829407 

230 

867862 

193 

961545 

423 

038455 

29 
i  30 

829545 

230 

867747 

193 

961799 

423 

038201 

31  > 

829683 

230 

867631 

193 

962052 

423 

037948 

30  ) 

)  31 

9829821 

229 

9-867515 

193 

9-962306 

423 

10037694 

29  j 

>  32 

829959 

229 

867399 

193 

962560 

423 

037440 

28  ( 

)  33 

830097 

229 

867283 

193 

962813 

423 

037187 

27  ( 

)  34 

830234 

229 

867167 

193 

963067 

423 

036933 

26  ( 

)  35 

830372 

229 

867051 

193 

963320 

423 

036680 

25  ( 

>  36 

830509 

229 

866935 

194 

963574 

423 

036426 

24  ( 

)  37 

830646 

229 

866819 

194 

963827 

423 

036173 

23  I 

/  38 

830784 

229 

866703 

194 

964081 

423 

035919 

22  ( 

)  39 

830921 

228 

866586 

194 

964335 

423 

035665 

21  ( 

(  40 

831058 

228 

866470 

194 

964588 

422 

035412 

20  I 

(  41 

9-831195 

228 

9-866353 

194 

9-964842 

422 

10035158 

19  J 

<  42 

831332 

228 

866237 

194 

965095 

422 

034905 

18  S 

{  43 

831469 

228 

866120 

194 

965349 

422 

034651 

17  > 

<  44 

831606 

228 

866004 

195 

965602 

422 

034398 

16  ) 

(  45 

831742 

,  228 

865887 

195 

965855 

422 

034145 

15  > 

{  46 

831879 

■  228 

865770 

195 

966109 

422 

033891 

14  ) 

(  47 

832015 

227 

865653 

195 

966362 

422 

033638 

13) 

(  48 

832152 

227 

865536 

195 

966616 

422 

033384 

12  ) 

(  49 

832288 

227 

865419 

195 

966869 

422 

033131 

11  ) 

J  50 

832425 

227 

865302 

195 

967123 

422 

032877 

10  j 

j  51 

9-832561 

227 

9'865185 

195 

9-967376 

422 

10032624 

9> 

)  52 

832697 

227 

865068 

195 

967629 

422 

032371 

8( 

)  53 

832833 

227 

864950 

195 

967883 

422 

032117 

7^ 

/  54 

)  55 
5  56 

832969 

226 

864833 

196 

968136 

422 

031864 

6> 

833105 

226 

864716 

196 

968389 

422 

031611 

5? 

833241 

226 

864598 

196 

968643 

422 

031357 

4  I 

)  57 

833377 

226 

864481 

196 

968896 

422 

031104 

3) 

58 

833512 

226 

864363 

196 

969149 

422 

030851 

2? 

)  59 

833648 

226 

864245 

196 

969403 

422 

030597 

J! 

|  60 

833783 

226 

864127 

196 

969656 

422 

030344  ' 

\\ 

Cosine! 

*~^J 

Sine    ! 

Cotang.  | 

^N^M«! 

^^Tang^l 

.M-J 

47  Degrees. 


SINES    AND    tangents.     (43  Degrees.) 


179 


M.   I 


I      Cosine 


Tang.         |       D.       |        Cotang.       | 


0 

9-833783 

226 

9-864127 

196 

9-969656 

422 

10-030344 

60^ 

1 

833919 

225 

864010 

196 

969909 

422 

030091 

59? 

2 

834054 

225 

863892 

197 

970162 

422 

029838 

58? 

3 

834189 

225 

863774 

197 

970416 

422 

029584 

57? 

4 

834325 

225 

863656 

197 

970669 

422 

029331 

56? 

5 

834400 

225 

863538 

197 

970922 

422 

029078 

55 

6 

834595 

225 

863419 

197 

971175 

422 

028825 

54? 

7 

834730 

225 

863301 

197 

971429 

422 

028571 

53 

8 

834865 

2-25 

863183 

197 

971682 

422 

028318 

52? 

9 

834999 

224 

863064 

197 

971935 

422 

028065 

51  ( 

10 

835134 

224 

862946 

198 

972188 

422 

02*812 

50? 

11 

9-835269 

224 

9-862827 

198 

9-972441 

422 

10027559 

49  j 

12 

835403 

224 

862709 

198 

972694 

422 

027306 

48  ( 

13 

835538 

224 

862590 

198 

972948 

422 

027052 

47  ( 

14 

835672 

224 

862471 

198 

973201 

422 

026799 

46  < 

15 

835807 

224 

862353 

198 

973454 

422 

026546 

45  ( 

16 

835941 

224 

862234 

198 

973707 

422 

026293 

44  ( 

17 

836075 

223 

862115 

198 

973960 

422 

026040 

43  ( 

18 

836209 

223 

861996 

198 

974213 

422 

025787 

42  ( 

19 

836343 

223 

861877 

198 

974466 

422 

025534 

41  C 

20 

836477 

223 

861758 

199 

974719 

422 

025281 

40  ( 

21 

9  836611 

223 

9-861638 

199 

9-974973 

422 

10025027 

39) 

22 

836745 

223 

861519 

199 

975220 

422 

024774 

38( 

23 

836878 

223 

861400 

199 

975479 

422 

024521 

37  < 

24 

837012 

222 

861280 

199 

975732 

422 

024268 

36  ) 

25 

837146 

222 

861161 

199 

975985 

422 

024015 

35  S 

26 

837279 

222 

861041 

199 

976238 

422 

023762 

34S 

27 

837412 

222 

860922 

199 

976491 

422 

023509 

33  S 

28 

837546 

222 

800802 

199 

976744 

422 

023256 

32< 

29 

837679 

222 

860682 

200 

976997 

422 

023003 

31  ( 

30 

837812 

222 

860562 

200 

977250 

422 

022750 

30  J 

31 

9-837945 

222 

9-860442 

200 

9*977503 

422 

10-022497 

29) 

32 

838078 

221 

860322 

200 

977756 

422 

022244 

28) 

33 

838211 

221 

860202 

200 

978009 

422 

021991 

27) 

34 

838344 

221 

860082 

200 

978262 

422 

021738 

26) 

35 

838477  ' 

221 

859962 

200 

978515 

422 

021485 

25) 

36 

8386 JO 

221 

859842 

200 

978768 

422 

021232 

24  ) 

37 

838742 

221 

859721 

201 

979021 

422 

020979 

23) 

38 

838875 

221 

859601 

201 

979274 

422 

020726 

22  ) 

39 

839007 

221 

859480 

201 

979527 

422 

020473 

21  ) 

40 

839140 

220 

859360 

201 

979780 

422 

020220 

20) 

41 

9-839272 

220 

9-859239 

201 

9-980033 

422 

10019967 

19  ) 

42 

839404 

220 

859119 

201 

980286 

422 

019714 

18) 

43 

839536 

220 

858998 

201 

980538 

422 

019462 

17  ) 

44 

839668 

220 

858877 

201 

980791 

421 

019209 

16) 

45 

839800 

220 

858756 

202 

981044 

421 

018956 

15) 

46 

839932 

220 

858635 

202 

981297 

421 

018703 

14  ) 

47 

840064 

219 

858514 

202 

981550 

421 

018450 

13  ) 

48 

840196 

219 

858393 

202 

981803 

421 

018197 

12  ) 

49 

840328 

219 

858272 

202 

982056 

421 

017944 

11  ) 

50 

840459 

219 

858151 

202 

982309 

421 

017691 

io  S 

51 

9-840591 

219 

9-858029 

202 

9-982562 

421 

10-017438 

9( 

52 

840722 

219 

857908 

202 

982814 

421 

017186 

8? 

53 

840854 

219 

857786 

202 

988067 

421 

016933 

7  > 

54 

840985 

219 

857665 

203 

983320 

421 

016680 

6) 

55 

841116 

218 

857543 

203 

983573 

421 

016427 

5  ) 

56 

841247 

218 

857422 

203 

983826 

421 

016174 

4  ) 

57 

841378 

218 

857300 

203 

984079 

421 

015921 

3? 

58 

841509 

218 

857178 

203 

984331 

421 

015669 

2) 

59 

841640 

218 

857056 

203 

984584 

421 

.  015416 

l) 

60 

841771 

218 

856934 

203 

984837 

421 

015163 

o) 

.X^S^l^J, 


46  Degrees. 


180 


(44 Degrees.)    a  table   of   logarithmic 


M.   | 


I      D. 


Tan?. 


|       Cotang.       | 


(  0 

9-841771 

ai8 

9-856934 

203 

1  9-984837 

421 

10015163 

60 

I   1 

841902 

218 

856812 

2U3 

985090 

421 

014910 

59 

<  2 

842033 

218 

856690 

204 

985343 

421 

014657 

58 

(  3 

842163 

217 

856568 

204 

985596 

421 

014404 

57 

(  4 

842294 

217 

856446 

204 

985848 

421 

014152 

56 

(  5 

842424 

217 

856323 

204 

986101 

421 

013899 

55 

(  6 

842555 

217 

856201 

204 

986354 

421 

013646 

54 

(  7 

842685 

217 

856078 

204 

986607 

421 

013393 

53 

(  8 

842815 

217 

855956 

204 

986860 

421 

013140 

52 

l    9 

842946 

217 

855833 

204 

987112 

421 

012888 

51 

J  10 

843076 

217 

855711 

205 

987365 

421 

012635 

50 

i  11 

9843206 

216 

9-855583 

205 

9-987618 

421 

10-012382 

49 

(  12 

843336 

216 

855465 

205 

987871 

421 

012129 

48 

(  13 

843466 

216 

855342 

205 

988123 

421 

011877 

47 

s  14 

843595 

216 

855219 

205 

988376 

421 

011624 

46 

S  15 

843725 

216 

855096 

205 

988629 

421 

011371 

45 

(  16 

843855 

216 

854973 

205 

988882 

421 

011118 

44 

I   I7 

843984 

216 

854850 

205 

989134 

421 

010866 

43 

(  18 

844114 

215 

854727 

206 

989387 

421 

010613 

42 

(  19 

844243 

215 

854603 

206 

989640 

421 

010360 

41 

(20 

844372 

215 

854480 

206 

989893 

421 

010107 

40  < 

(21 

9-844502 

215 

9854356 

206 

9-990145 

421 

10009855 

39 

S  22 

844631 

215 

854233 

206 

.  990398 

421 

009602 

38 

<  23 

844760 

215 

854109 

206 

990651 

421 

009349 

37 

<  24 

844889 

215 

853986 

206 

990903 

421 

009097 

36 

<25 

845018 

215 

853862 

206 

991156 

421 

008844 

35 

<  26 

845147 

215 

853738 

206 

991409 

421 

008591 

34 

(  27 

845276 

214 

853614 

207 

991662 

421 

008338 

33 

<  28 

845405 

214 

853490 

207 

991914 

421 

008086 

32 

<  29 

845533 

214 

853366 

207 

992167 

421 

007833 

31 

(  30 

845662 

214 

853242 

207 

992420 

421 

007580 

30 

)  31 

9-845790 

214 

9-853118 

207 

9-992672 

421 

10-007328 

29 

)  32 

845919 

214 

852994 

207 

992925 

421 

007075 

28 

)  33 

846047 

214 

852869 

207 

993178 

421 

006822 

27 

)  34 

846175 

214 

852745 

207 

993430 

421 

006570 

26 

)  35 

846304 

214 

852620 

207 

993683 

421 

006317 

25 

)  36 

846432 

213 

852496 

208 

993936 

421 

006064 

24 

)  37 

846560 

213 

852371 

208 

994189 

421 

005811 

23 

S  38 

846688 

213 

852247 

208 

994441 

421 

005559 

22 

S  39 

846816 

213 

852122 

208 

994694 

421 

005306 

21 

S  40 

846944 

213 

851997 

208 

994947 

421 

005053 

20) 

j  41 

9-847071 

213 

9-851872 

208 

9-995199 

421 

10004801 

lo) 

>  42 

847199 

213 

851747 

208 

995452 

421 

004548 

18  P 

)  43 

847327 

213 

851622 

208 

995705 

421 

004295 

n) 

)  44 

847454 

212 

851497 

209 

995957 

421 

004043 

16? 

)  45 

847582 

212 

851372 

209 

996210 

421 

003790 

15  ( 

)  46 

847709 

212 

851246 

209 

996463 

421 

003537 

14  ( 

)  47 

847836 

212 

851121 

209 

996715 

421 

003285 

13/ 

)  48 

847964 

212 

850996 

209  . 

996968 

421 

003032 

12/ 

)  49 

848091 

212 

850870 

209 

997221 

421 

002779 

11  / 

)  50 

848218 

212 

850745 

209 

997473 

421 

002527 

10  \ 

<51 

9-848345 

212 

9-850619 

209 

9-997726 

421 

10002274 

9< 

(52 

848472 

211 

850493 

210 

997979 

421 

002021 

8< 

(53 

848599 

211 

850368 

210 

998231 

421 

001769 

7  ( 

<54 

848726 

211 

850242 

210 

998484 

421 

001516 

6S 

(  55 

848852 

211 

850116 

210 

998737 

421 

001263 

5  ( 

<  56 

•  848979 

211 

849990 

210 

998989 

421 

001011 

4S 

<  57 

849106 

211 

849864 

210 

999242 

421 

000758 

3S 

?  58 

849232 

211 

849738 

210 

999495 

421 

000505 

2( 

<  59 

849359 

211 

849611 

210 

999748 

421 

000253 

1  ( 

I  00 

849485 

211 

849485 

210 

10000000 

421 

000000 

0  ( 

I         Tang.       I   M.' 


45  Degrees 


THE  La 
OW 


1      RETURN    CIRCULATION  DEPARTMENT 

TO— ►    202  Main  Library 

;       LOAN  PERIOD  1 
HOME  USE 

2 

3 

4 

5 

5 

ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS 

Renewals  and  Recharges  may  be  made  4  days  prior  to  the  due  date. 

Books  may  be  Renewed  by  calling        642-3405 

DUE  AS  STAMPED  BELOW 

RECEIVED 

JUN  2  9  1996 

CIRCULATION  DEP 

t 

UNIVERSITY  OF  CALIFORNIA,  BERKELEY 
FORM  NO.  DD6                                 BERKELEY,  CA  94720             @$ 

U.  C.  BERKELEY  L180.^.?.. 

CD571EES22 


V? 


: 

ffZLW 

i 

Qasi 

B 

THE  UNIVERSITY  OF  CALIFORtflX^BRARY 


-    ^      * 


%f.  ■'**• 


